Simple derivation of general Fierz-type identities

American Journal of Physics

Volumn 73, Issue 12, 2005, Pages 1160-1163

  Nishi, C C ^a  

^a SÃO PAULO STATE UNIVERSITY UNESP   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 30344481374     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.2074087     Document Type: Article

References (18)

1. M. Fierz, "Zur Fermischen Theorie des β-Zerfalls," Z. Phys. 104, 553-565 (1937).
2. J. F. Donoghue, E. Golowich, and B. R. Holstein, Dynamics of the Standard Model (Cambridge University Press, 1994), pp. 217 and 221.
3. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1965).
4. Y. Takahashi, "The Fierz identities," in Progress in Quantum Field Theory, edited by H. Ezawa and S. Kamefuchi (North-Holland, Amsterdam, 1986), p. 121.
5. J. F. Nieves and P. B. Pal, "Generalized Fierz identities," Am. J. Phys. 72, 1100-1108 (2004).
6. R. Mohapatra and P. Pal, Massive Neutrinos in Physics and Astrophysics (World Scientific, Singapore, 1991), pp. 165-70.
7. The surprising implications and experimental evidences for neutrino oscillations can be found in W. C. Haxton and B. R. Holstein, "Neutrino physics," Am. J. Phys. 68, 15-32 (2000);
8. also W. C. Haxton and B. R. Holstein, "Neutrino physics: An update," Am, J. Phys. 72, 18-24 (2004);
9. see also M. C. Gonzalez-Garcia and Y. Nir, "Neutrino masses and mixing: Evidence and implications," Rev. Mod. Phys. 75, 345-402 (2003).
10. See any linear algebra book, for example, I. M. Gel'fand, Lectures on Linear Algebra (Interscience, New York, 1961).
11. Any operation applied to an element in the vector space must result in another element of the vector space.
12. 2 purely complex matrices with real expansion coefficients only, that is, over the reals R.
13. See C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980), p. 516, for an explicit representation of Gell-Mann matrices.
14. a}, respectively.
15. I will denote as Dirac bilinears the proper bilinears containing the two spinors as well as the associated matrices alone, because the Fierz identities do not depend on the spinors involved.
16. The lowering of space-time indices is equivalent to the hermitian conjugation operation.
17. See Ref. 11, Appendix.
18. These identities were used to get from Eq. (1) to Eq. (2) with a sign difference due to the anticommutation of fermion fields.