Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 021, 46 pages      arXiv:0802.2634      https://doi.org/10.3842/SIGMA.2008.021
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices

Victor M. Red'kov, Andrei A. Bogush and Natalia G. Tokarevskaya
B.I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Minsk, Belarus

Received September 19, 2007, in final form January 24, 2008; Published online February 19, 2008

Abstract
Parametrization of 4 × 4-matrices G of the complex linear group GL(4,C) in terms of four complex 4-vector parameters (k,m,n,l) is investigated. Additional restrictions separating some subgroups of GL(4,C) are given explicitly. In the given parametrization, the problem of inverting any 4 × 4 matrix G is solved. Expression for determinant of any matrix G is found: det G = F(k,m,n,l). Unitarity conditions G+ = G-1 have been formulated in the form of non-linear cubic algebraic equations including complex conjugation. Several simplest solutions of these unitarity equations have been found: three 2-parametric subgroups G1, G2, G3 - each of subgroups consists of two commuting Abelian unitary groups; 4-parametric unitary subgroup consisting of a product of a 3-parametric group isomorphic SU(2) and 1-parametric Abelian group. The Dirac basis of generators Λk, being of Gell-Mann type, substantially differs from the basis λi used in the literature on SU(4) group, formulas relating them are found - they permit to separate SU(3) subgroup in SU(4). Special way to list 15 Dirac generators of GL(4,C) can be used {Λk} = {αiÅβjÅ(αiVβj = KÅL ÅM )}, which permit to factorize SU(4) transformations according to S = eiaα eibβeikKeilLeimM, where two first factors commute with each other and are isomorphic to SU(2) group, the three last ones are 3-parametric groups, each of them consisting of three Abelian commuting unitary subgroups. Besides, the structure of fifteen Dirac matrices Λk permits to separate twenty 3-parametric subgroups in SU(4) isomorphic to SU(2); those subgroups might be used as bigger elementary blocks in constructing of a general transformation SU(4). It is shown how one can specify the present approach for the pseudounitary group SU(2,2) and SU(3,1).

Key words: Dirac matrices; linear group; unitary group; Gell-Mann basis; parametrization.

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References

  1. Baker H.F., On the exponential theorem for a simply transitive continuous group, and the calculation of the finite equations from the constants of structure, Proc. London Math. Soc. 34 (1901), 91-129.
    Baker H.F., Further applications of matrix notation to integration problems, Proc. London Math. Soc. 34 (1901), 347-360.
  2. Barenco A., A universal two bit gate for quantum computation, Proc. Roy. Soc. London A 449 (1995), 679-693.
  3. Barnes K.J., Delbourgo R., Matrix and tensor constructions from a generic SU(n) vector, J. Phys. A: Math. Gen. 5 (1972), 1043-1053.
  4. Barnes K.J., Dondi P.H., Sarkar S.C., General form of the SU(3) Gursey matrix, J. Phys. A: Math. Gen. 5 (1972), 555-562.
  5. Barut A.O., Bohm A., Reduction of a class of O(4,2) representations with respect to SO(4,1) and SO(3,2), J. Math. Phys. 11 (1970), 2938-2945.
  6. Barut A.O., Bracken A.J., Zitterbewegung and the internal geometry of the electron, Phys. Rev. D 23 (1981), 2454-2463.
  7. Barut A.O., Brittin W.E., De Sitter and conformal groups and their applications, Lecture Notes in Theoretical Physics, Vol. 13, 1971.
  8. Barut A.O., Zeni J.R., Laufer A.J., The exponential map for the conformal group O(2,4), J. Phys. A: Math. Gen. 27 (1994), 5239-5250, hep-th/9408105. 1994.
  9. Barut A.O., Zeni J.R., Laufer A., The exponential map for the unitary group SU(2,2), J. Phys. A: Math. Gen. 27 (1994), 6799-6806, hep-th/9408145.
  10. Bég M.A.B., Ruegg H., A set of harmonic functions for the group SU(3), J. Math. Phys. 6 (1965), 677-682.
  11. Bincer A.M., Parametrization of SU(n) with n-1 orthonormal vectors, J. Math. Phys. 31 (1990), 563-567.
  12. Bogush A.A., On matrices of finite unitary transformations, Vesti AN BSSR Ser. Fiz.-Mat. (1973), no. 5, 105-112 (in Russian).
  13. Bogush A.A., On subgroups of the group SU(n) isomorphic to SU(2) and flat unitary transformations, Vesti AN BSSR Ser. Fiz.-Mat. (1974), no. 1, 63-68 (in Russian).
  14. Bogush A.A., On vector parameterization of subgroups of the group SO(n,C) isomorphic to SO(3,C), Doklady AN BSSR 17 (1973), 995-999 (in Russian).
  15. Bogush A.A., Fedorov F.I., On flat orthogonal transformations, Doklady AN SSSR 206 (1972), 1033-1036 (in Russian).
  16. Bogush A.A., Fedorov F.I., Fedorovykh A.M., On finite transformations of the group SO(n,R) and its representations, Doklady AN SSSR 214 (1974), 985-988 (in Russian).
  17. Bogush A.A., Otchik V.S., Fedorov F.I., On finite transformations of the group SO(5) and its finite-dimensional representations, Doklady AN SSSR 227 (1976), 265-268 (in Russian).
  18. Bogush A.A., Red'kov V.M., On vector parametrization of the group GL(4,C) and its subgroups, Vesti National Academy of Sciences of Belarus, Ser Fiz.-Mat. (2006), no. 3, 63-69.
    Bogush A.A., Red'kov V.M., On unique parametrization of the linear group GL(4,C) and its subgroups by using the Dirac matrix algebra basis, hep-th/0607054.
  19. Bracken A.J., Massive neutrinos, massless neutrinos, and so(4,2) invariance, hep-th/0504111.
  20. Bracken A.J., O(4,2): an exact invariance algebra for the electron, J. Phys. A: Math. Gen. 8 (1975), 808-815.
  21. Bracken A.J., A simplified SO(6,2) model of SU(3), Comm. Math. Phys. 94 (1984), 371-377.
  22. Byrd M., Differential geometry on SU(3) with applications to three state systems, J. Math. Phys. 39 (1998), 6125-6136, Erratum, J. Math. Phys. 41 (2001), 1026-1030, math-ph/9807032.
  23. Byrd M., Geometric phases for three state systems, quant-ph/9902061.
  24. Byrd M., The geometry of SU(3), physics/9708015.
  25. Byrd M., Sudarshan E.C.G., SU(3) revisited, J. Phys. A: Math. Gen. 31 (1998), 9255-9268, physics/9803029.
  26. Campbell J.E., On a law of combination of operators bearing on the theory of continuous transformation groups, Proc. London Math. Soc. 28 (1896), 381-390.
    Campbell J.E., On a law of combination of operators (second paper), Proc. London Math. Soc. 29 (1897), 14-32.
  27. Chacon E., Moshinski M., Representations of finite U(3) transformations, Phys. Lett. 23 (1966), 567-569.
  28. Chaturvedi S., Mukunda N., Parametrizing the mixing matrix: a unified approach, Internat. J. Modern Phys. A 16 (2001), 1481-1490, hep-ph/0004219.
  29. Dahiya H., Gupta M., SU(4) chiral quark model with configuration mixing, Phys. Rev. D 67 (2003), 074001, 8 pages, hep-ph/0302042.
  30. Dahm R., Relativistic SU(4) and quaternions, Adv. Appl. Clifford Algebras 7 (1997), suppl., 337-356, hep-ph/9601207.
  31. Deutsch D., Quantum computational networks, Proc. Roy. Soc. London. A 425 (1989), 73-90.
  32. Fedorov F.I., The Lorentz group, Nauka, Moscow, 1979.
  33. Fedorov F.I., Fedorovykh A.V., Covariant parameterization of the group SU(3), Vesti AN BSSR Ser. Fiz.-Mat. (1975), no. 6, 42-46 (in Russian).
  34. González P., Vijande J., Valcarce A., Garcilazo H., A SU(4)ÄO(3) scheme for nonstrange baryons, Eur. Phys. J. A 31 (2007), 515-518, hep-ph/0610257.
  35. Gsponer A., Explicit closed form parametrization of SU(3) and SU(4) in terms of complex quaternions and elementary functions, math-ph/0211056.
  36. Guidry M., Wu L.-A., Sun Y., Wu C.-L., An SU(4) model of high-temperature superconductivity and antiferromagnetism, Phys. Rev. B 63 (2001), 134516, 11 pages, cond-mat/0012088.
  37. Hausdorff F., Untersuchungen über Ordnungstypen: I, II, III. Ber. über die Verhandlungen der Königl. Sächs. Ges. der Wiss. zu Leipzig. Math.-phys. Klasse 58 (1906), 106-169.
    Hausdorff F., Untersuchungen über Ordnungstypen: IV, V, Ber. über die Verhandlungen der Königl. Sächs. Ges. der Wiss. zu Leipzig. Math.-phys. Klasse 59 (1907), 84-159.
  38. Holland D.F., Finite transformations of SU(3), J. Math. Phys. 10 (1969), 531-535.
  39. Keane A.J., Barrett R.K., The conformal group SO(4,2) and Robertson-Walker spacetimes, Classical Quantum Gravity 17 (2000), 201-218, gr-qc/9907002.
  40. Kihlberg A., Müller V.F., Halbwachs F., Unitary irreducible representations of SU(2,2), Gomm. Math. Phys. 3 (1966), 194-217.
  41. Kleefeld F., Dillig M., Trace evaluation of matrix determinants and inversion of 4 × 4 matrices in terms of Dirac covariants, hep-ph/9806415.
  42. Kusnezov D., Exact matrix expansions for group elements of SU(N), J. Math. Phys. 36 (1995), 898-906.
  43. Lanczos C., Linear systems in selfadjoint form, Amer. Math. Monthly 65 (1958), 665-679, reprinted and commented Cornelius Lanczos Collected Published Papers with Commentaries, Editors W.R. Davis et al., North Carolina State University, Raleigh, 1998, Vol. V, 3191-3205.
  44. Macfarlane A.J., Description of the symmetry group SU3/Z3 of the octet model, Comm. Math. Phys. 11 (1968), 91-98.
  45. Macfarlane A.J., Dirac matrices and the Dirac matrix description of Lorentz transformations, Comm.Math. Phys. 2 (1966), 133-146.
  46. Macfarlane A.J., Parametrizations of unitary matrices and related coset spaces, J. Math. Phys. 21 (1980), 2579-2582.
  47. Macfarlane A.J., Sudbery A., Weisz P.H., On Gell-Mann's l-matrices, d- and f-tensors, octets, and parametrizations of SU(3), Comm. Math. Phys. 11 (1968), 77-90.
  48. Mack G., All unitary ray representations of the conformal group SU(2,2) with positive energy, Comm. Math. Phys. 55 (1977), 1-28.
  49. Mack G., Salam A., Finite-component field representations of the conformal group, Ann. Phys. 53 (1969), 174-202.
  50. Mack G., Todorov I.T., Irreducibility of the ladder representations of U(2,2) when restricted to the Poincaré group, J. Math. Phys. 10 (1969), 2078-2085.
  51. Marchuk N.G., A model of the composite structure of quarks and leptons with SU(4) gauge symmetry, hep-ph/9801382.
  52. Mishra A., Ma M., Zhang F.-C., Plaquette ordering in SU(4) antiferromagnets, Phys. Rev. B 65 (2002), 214411, 6 pages, cond-mat/0202132.
  53. Moler C., Van Loan C., Nineteen dubious ways to compute the exponential of a matrix, SIAM Rev. 20 (1978), 801-836.
  54. Moshinsky M., Wigner coefficients for the SU3 group and some applications, Rev. Modern Phys. 34 (1962), 813-828.
  55. Mukunda N., Arvind, Chaturvedi S., Simon R., Bargmann invariants and off-diagonal geometric phases for multi-level quantum systems - a unitary group approach, Phys. Rev. A 65 (2003), 012102, 10 pages, quant-ph/0107006.
  56. Murnaghan F.D., The unitary and rotation group, Spartan Books, Washington, 1962.
  57. Nelson T.J., A set of harmonic functions for the group SU(3) as specialized matrix elements of a general finite transformation, J. Math. Phys. 8 (1967), 857-863.
  58. Olevskiy M.N., Three-orthogonal systems in spaces of constant curvature in which equation D2U + lU = 0 permits the full separation of variables, Mat. Sb. 27 (1950), 379-426 (in Russian).
  59. Raghunathan K., Seetharaman M., Vasan S.S., A disentanglement relation for SU(3) coherent states, J. Phys. A: Math. Gen. 22 (1989), L1089-L1092.
  60. Ramakrishna V., Costa F., On the exponentials of some structured matrices, J. Phys A: Math. Gen. 37 (2004), 11613-11628, math-ph/0407042.
  61. Ramakrishna V., Zhou H., On the exponential of matrices in su(4), math-ph/0508018.
  62. Red'kov V.M., Bogush A.A., Tokarevskaya N.G., 4 × 4 matrices in Dirac parametrization: inversion problem and determinant, arXiv:0709.3574.
  63. Rosen S.P., Finite transformations in various representations of SU(3), J. Math. Phys. 12 (1971), 673-681.
  64. Sánchez-Monroy J.A., Quimbay C.J., SU(3) Maxwell equations and the classical chromodynamics, hep-th/0607203.
  65. Schirmer S.G., Greentree A.D., Ramakrishna V., Rabitz H., Constructive control of quantum systems using factorization of unitary operators, J. Phys. A: Math. Gen. 35 (2002), 8315-8339, quant-ph/0211042.
  66. Ten Kate A., Dirac algebra and a six-dimensional Lorentz group, J. Math. Phys. 9 (1968), 181-185.
  67. Terazawa H., Chikashige Y., Akama K., Unified model of the Nambu-Jona-Lasinio type for all elementary-particle forces, Phys. Rev. D. 15 (1977), 480-487.
  68. Terazawa Y., Subquark model of leptons and quarks, Phys. Rev. D 22 (1980), 184-199.
  69. Tilma T., Byrd M., Sudarshan E.C.G., A parametrization of bipartite systems based on SU(4) Euler angles, J. Phys. A: Math. Gen. 35 (2002), 10445-10465, math-ph/0202002.
  70. Tilma T., Sudarshan E.C.G., Generalized Euler angle parametrization for SU(N), J. Phys. A: Math. Gen. 35 (2002), 10467-10501, math-ph/0205016.
  71. Tilma T., Sudarshan E.C.G., Generalized Euler angle parameterization for U(N) with applications to SU(N) coset volume measures, J. Geom. Phys. 52 (2004), 263-283, math-ph/0210057.
  72. Vlasov A.Yu., Dirac spinors and representations of GL(4) group in GR, math-ph/0304006.
  73. Volovik G.E., Dark matter from SU(4) model, JETP Lett. 78 (2003), 691-694, hep-ph/0310006.
  74. Weigert S., Baker-Campbell-Hausdorff relation for special unitary groups SU(n), J. Phys. A: Math. Gen. 30 (1997), 8739-8749, quant-ph/9710024.
  75. Wilcox R.M., Exponential operators and parameter differentiation in quantum physics, J. Math. Phys. 8 (1967), 962-982.

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