Abstract
In this chapter we present the basics of Dirac spinor algebra needed for calculations involving fermions. We introduce the commuting and anti-commuting relations among the various Dirac matrices and we present the basics of calculating spinor traces. The tools given here can be used to further calculate lengthy and more complicated traces involving Dirac matrices. The transformation of spinors under Lorentz transformations is also presented consistently together with the bilinear covariants. Finally a short comparison between QED and QCD is given for a simple process.
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Notes
- 1.
See Chap. 1 for the definition of tensor density.
- 2.
For a nice review of how this is done read A. Pich, The Standard Model of Electroweak Interactions, http://arxiv.org/pdf/1201.0537v1.pdf and A. Pich, Quantum Chromodynamics, http://arxiv.org/pdf/hep-ph/9505231.pdf.
- 3.
For more details one should consult the last four references at the end of this chapter.
- 4.
For a complete set of Feynman rules for QCD (and the SM in general) see J. C. Romao and J. P. Silva, A resource for signs and Feynman diagrams of the Standard Model, Int. J. Mod. Phys. A 27 (2012) 1230025, http://arxiv.org/pdf/1209.6213.pdf
- 5.
The result from Peskin and Schroeder for this cross section seems not to be correct, however our result agrees (except a 1/2 factor which they did not include) with R.K. Ellis, W.J. Stirling, B.R. Webber, QCD and Collider Physics.
Further Reading
A. Pich, Quantum Chromodynamics, http://arxiv.org/pdf/hep-ph/9505231.pdf
G. Dissertori, I.G. Knowles, M. Schmelling, Quantum Chromodynamics
A. Pich, The Standard Model of Electroweak Interactions, http://arxiv.org/pdf/1201.0537v1.pdf
T. Muta, Foundations of Quantum Chromodynamics
J.C. Romao, J.P. Silva, A resource for signs and Feynman diagrams of the standard model, Int. J. Mod. Phys. A 27 (2012) 1230025, http://arxiv.org/pdf/1209.6213.pdf
R.K. Ellis, W.J. Stirling, B.R. Webber, QCD and Collider Physics
T.P. Cheng, L.F. Li, Gauge Theory of Elementary Particle Physics, (Oxford, 1984)
L.H. Ryder, Quantum Field Theory, (Cambridge University Press, 1985)
F. Mandl, G.P. Shaw, Quantum Field Theory
M. Kaku, Quantum Field Theory: A Modern Introduction.
M.E. Peskin, D.V. Schroeder, An Introduction To Quantum Field Theory (Addison-Wesley Publishing Company, San Francisco, 1995)
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Ilisie, V. (2016). Dirac Algebra. In: Concepts in Quantum Field Theory. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-22966-9_5
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DOI: https://doi.org/10.1007/978-3-319-22966-9_5
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