Skip to main content
SpringerLink
Log in

Geometric Invariant Decomposition of \(\text {SU}(\textbf{3})\)

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

A novel invariant decomposition of diagonalizable \(n \times n\) matrices into n commuting matrices is presented. This decomposition is subsequently used to split the fundamental representation of \(\mathfrak {su}({3})\) Lie algebra elements into at most three commuting elements of \(\mathfrak {u}({3})\). As a result, the exponential of an \(\mathfrak {su}({3})\) Lie algebra element can be split into three commuting generalized Euler’s formulas, or conversely, a Lie group element can be factorized into at most three generalized Euler’s formulas. After the factorization has been performed, the logarithm follows immediately.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability statement

Data sharing not applicable.

Notes

  1. The same logic is also valid for the complex logarithm of a complex number \(U = e^{i \theta }\), if the complex logarithm is defined as \({{\,\textrm{Ln}\,}}U = \widehat{\langle U\rangle }_2 \arccos (\langle U\rangle _0)\), where \(\langle U\rangle _0 = \tfrac{1}{2} (U + U^*)\) and \(\langle U\rangle _2 = \tfrac{1}{2} (U - U^*)\). This construction allows us to maintain the distinction between \(\pm U\), without requiring us to first distinguish the appropriate quadrant of the complex plane before computing \(\arctan \), such as is typically done in complex analysis.

References

  1. Ablowitz, M. J., Fokas, A. S.: Complex Variables: Introduction and Applications. 2nd ed. Cambridge Texts in Applied Mathematics. Cambridge University Press, (2003). https://doi.org/10.1017/CBO9780511791246.

  2. Curtright, T.L., Zachos, C.K.: Elementary results for the fundamental representation of SU(3). Rept. Math. Phys. 76, 401–404 (2015). https://doi.org/10.1016/S0034-4877(15)30040-9

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. DeGrand, T., DeTar, C.: Lattice Methods for Quantum Chromodynamics (2006). https://doi.org/10.1142/6065

  4. Doran, C., et al.: Lie groups as spin groups. J. Math. Phys. 34(8), 3642–3669 (1993). https://doi.org/10.1063/1.530050

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Gell-Mann, M.: “The Eightfold Way: A Theory of strong interaction symmetry”. In: (Mar. 1961). https://doi.org/10.2172/4008239.

  6. Giusti, L., et al.: Problems on lattice gauge fixing. Int. J. Mod. Phys. A 16, 3487–3534 (2001). https://doi.org/10.1142/S0217751X01004281

    Article  ADS  MATH  Google Scholar 

  7. Hestenes, D., Sobczyk, G.: Clifford algebra to geometric calculus : a unified language for mathematics and physics. Dordrecht; Boston; Hing- ham, MA, U.S.A.: D. Reidel ; Distributed in the U.S.A. and Canada by Kluwer Academic Publishers (1984)

  8. Mandula, J.E., Ogilvie, M.: The gluon is massive: a lattice calculation of the gluon propagator in the Landau Gauge. Phys. Lett. B 185, 127–132 (1987). https://doi.org/10.1016/0370-2693(87)91541-3

    Article  ADS  Google Scholar 

  9. Peskin, M. E., Schroeder, D. V.: An Introduction To Quantum Field Theory. Frontiers in Physics. Avalon Publishing, (1995). isbn: 9780813345437

  10. Roelfs, M., De Keninck, S.: “Graded Symmetry Groups: Plane and Simple”. In: (under review at AACA). arXiv: 2107.03771

  11. Van Kortryk, T.S.: Matrix exponentials, SU(N) group elements, and real polynomial roots. J. Math. Phys. 57(2), 021701 (2016). https://doi.org/10.1063/1.4938418

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank Prof. David Dudal, Prof. Anthony Lasenby, and Steven De Keninck for valuable discussions about this research. Additional gratitude goes to the insights provided by geometric algebra, and [4, 7] in particular, which were the driving force behind this research. The research of M. R. was supported by KU Leuven IF project C14/16/067.

Author information

Authors and Affiliations

  1. Department of Physics, KU Leuven Campus Kortrijk–Kulak, Etienne Sabbelaan 53 bus 7657, 8500, Kortrijk, Belgium

    Martin Roelfs

Authors
  1. Martin Roelfs
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Roelfs.

Ethics declarations

Conflict of interest

This work does not have any conflicts of interest.

Additional information

Communicated by Uwe Kaehler.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Roelfs, M. Geometric Invariant Decomposition of \(\text {SU}(\textbf{3})\). Adv. Appl. Clifford Algebras 33, 5 (2023). https://doi.org/10.1007/s00006-022-01252-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-022-01252-w

Search

Navigation