Generalized Fierz identities

American Journal of Physics

Volumn 72, Issue 8, 2004, Pages 1100-1108

  Nieves, José F ^a   Pal, Palash B ^b  

^a UNIVERSITY OF PUERTO RICO   (Puerto Rico)

^b SAHA INSTITUTE OF NUCLEAR PHYSICS   (India)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 4344627797     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1757445     Document Type: Article

References (19)

1. Dirac matrices and spinors can be found in any book on quantum mechanics or quantum field theory. For example, see L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York 1968); A. Messiah, Quantum Mechanics (Dover, New York, 2000); L. H. Ryder, Quantum Field Theory, 2nd ed. (Cambridge U.P., Cambridge, 1996); Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory (Westview, 1995); Amitabha Lahiri and Palash B. Pal, A First Book of Quantum Field Theory (Narosa, 2000). In addition, see also R. E. Marshak, C. P. Ryan, and Riazuddin, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969), p. 82; C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw Hill, New York, 1980), p. 160; Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon, Oxford, 1984), p. 495; P. Renton, Electroweak Interactions (Cambridge U.P., Cambridge, 1990), p. 578; R. N. Mohapatra and P. B. Pal, Massive Neutrinos in Physics and Astrophysics, 2nd ed. (World Scientific, Singapore, 1998), p. 7.
2. Dirac matrices and spinors can be found in any book on quantum mechanics or quantum field theory. For example, see L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York 1968); A. Messiah, Quantum Mechanics (Dover, New York, 2000); L. H. Ryder, Quantum Field Theory, 2nd ed. (Cambridge U.P., Cambridge, 1996); Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory (Westview, 1995); Amitabha Lahiri and Palash B. Pal, A First Book of Quantum Field Theory (Narosa, 2000). In addition, see also R. E. Marshak, C. P. Ryan, and Riazuddin, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969), p. 82; C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw Hill, New York, 1980), p. 160; Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon, Oxford, 1984), p. 495; P. Renton, Electroweak Interactions (Cambridge U.P., Cambridge, 1990), p. 578; R. N. Mohapatra and P. B. Pal, Massive Neutrinos in Physics and Astrophysics, 2nd ed. (World Scientific, Singapore, 1998), p. 7.
3. Dirac matrices and spinors can be found in any book on quantum mechanics or quantum field theory. For example, see L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York 1968); A. Messiah, Quantum Mechanics (Dover, New York, 2000); L. H. Ryder, Quantum Field Theory, 2nd ed. (Cambridge U.P., Cambridge, 1996); Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory (Westview, 1995); Amitabha Lahiri and Palash B. Pal, A First Book of Quantum Field Theory (Narosa, 2000). In addition, see also R. E. Marshak, C. P. Ryan, and Riazuddin, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969), p. 82; C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw Hill, New York, 1980), p. 160; Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon, Oxford, 1984), p. 495; P. Renton, Electroweak Interactions (Cambridge U.P., Cambridge, 1990), p. 578; R. N. Mohapatra and P. B. Pal, Massive Neutrinos in Physics and Astrophysics, 2nd ed. (World Scientific, Singapore, 1998), p. 7.
4. Dirac matrices and spinors can be found in any book on quantum mechanics or quantum field theory. For example, see L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York 1968); A. Messiah, Quantum Mechanics (Dover, New York, 2000); L. H. Ryder, Quantum Field Theory, 2nd ed. (Cambridge U.P., Cambridge, 1996); Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory (Westview, 1995); Amitabha Lahiri and Palash B. Pal, A First Book of Quantum Field Theory (Narosa, 2000). In addition, see also R. E. Marshak, C. P. Ryan, and Riazuddin, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969), p. 82; C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw Hill, New York, 1980), p. 160; Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon, Oxford, 1984), p. 495; P. Renton, Electroweak Interactions (Cambridge U.P., Cambridge, 1990), p. 578; R. N. Mohapatra and P. B. Pal, Massive Neutrinos in Physics and Astrophysics, 2nd ed. (World Scientific, Singapore, 1998), p. 7.
5. Dirac matrices and spinors can be found in any book on quantum mechanics or quantum field theory. For example, see L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York 1968); A. Messiah, Quantum Mechanics (Dover, New York, 2000); L. H. Ryder, Quantum Field Theory, 2nd ed. (Cambridge U.P., Cambridge, 1996); Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory (Westview, 1995); Amitabha Lahiri and Palash B. Pal, A First Book of Quantum Field Theory (Narosa, 2000). In addition, see also R. E. Marshak, C. P. Ryan, and Riazuddin, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969), p. 82; C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw Hill, New York, 1980), p. 160; Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon, Oxford, 1984), p. 495; P. Renton, Electroweak Interactions (Cambridge U.P., Cambridge, 1990), p. 578; R. N. Mohapatra and P. B. Pal, Massive Neutrinos in Physics and Astrophysics, 2nd ed. (World Scientific, Singapore, 1998), p. 7.
6. Dirac matrices and spinors can be found in any book on quantum mechanics or quantum field theory. For example, see L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York 1968); A. Messiah, Quantum Mechanics (Dover, New York, 2000); L. H. Ryder, Quantum Field Theory, 2nd ed. (Cambridge U.P., Cambridge, 1996); Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory (Westview, 1995); Amitabha Lahiri and Palash B. Pal, A First Book of Quantum Field Theory (Narosa, 2000). In addition, see also R. E. Marshak, C. P. Ryan, and Riazuddin, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969), p. 82; C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw Hill, New York, 1980), p. 160; Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon, Oxford, 1984), p. 495; P. Renton, Electroweak Interactions (Cambridge U.P., Cambridge, 1990), p. 578; R. N. Mohapatra and P. B. Pal, Massive Neutrinos in Physics and Astrophysics, 2nd ed. (World Scientific, Singapore, 1998), p. 7.
7. Dirac matrices and spinors can be found in any book on quantum mechanics or quantum field theory. For example, see L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York 1968); A. Messiah, Quantum Mechanics (Dover, New York, 2000); L. H. Ryder, Quantum Field Theory, 2nd ed. (Cambridge U.P., Cambridge, 1996); Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory (Westview, 1995); Amitabha Lahiri and Palash B. Pal, A First Book of Quantum Field Theory (Narosa, 2000). In addition, see also R. E. Marshak, C. P. Ryan, and Riazuddin, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969), p. 82; C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw Hill, New York, 1980), p. 160; Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon, Oxford, 1984), p. 495; P. Renton, Electroweak Interactions (Cambridge U.P., Cambridge, 1990), p. 578; R. N. Mohapatra and P. B. Pal, Massive Neutrinos in Physics and Astrophysics, 2nd ed. (World Scientific, Singapore, 1998), p. 7.
8. Dirac matrices and spinors can be found in any book on quantum mechanics or quantum field theory. For example, see L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York 1968); A. Messiah, Quantum Mechanics (Dover, New York, 2000); L. H. Ryder, Quantum Field Theory, 2nd ed. (Cambridge U.P., Cambridge, 1996); Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory (Westview, 1995); Amitabha Lahiri and Palash B. Pal, A First Book of Quantum Field Theory (Narosa, 2000). In addition, see also R. E. Marshak, C. P. Ryan, and Riazuddin, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969), p. 82; C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw Hill, New York, 1980), p. 160; Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon, Oxford, 1984), p. 495; P. Renton, Electroweak Interactions (Cambridge U.P., Cambridge, 1990), p. 578; R. N. Mohapatra and P. B. Pal, Massive Neutrinos in Physics and Astrophysics, 2nd ed. (World Scientific, Singapore, 1998), p. 7.
9. Dirac matrices and spinors can be found in any book on quantum mechanics or quantum field theory. For example, see L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York 1968); A. Messiah, Quantum Mechanics (Dover, New York, 2000); L. H. Ryder, Quantum Field Theory, 2nd ed. (Cambridge U.P., Cambridge, 1996); Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory (Westview, 1995); Amitabha Lahiri and Palash B. Pal, A First Book of Quantum Field Theory (Narosa, 2000). In addition, see also R. E. Marshak, C. P. Ryan, and Riazuddin, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969), p. 82; C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw Hill, New York, 1980), p. 160; Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon, Oxford, 1984), p. 495; P. Renton, Electroweak Interactions (Cambridge U.P., Cambridge, 1990), p. 578; R. N. Mohapatra and P. B. Pal, Massive Neutrinos in Physics and Astrophysics, 2nd ed. (World Scientific, Singapore, 1998), p. 7.
10. Dirac matrices and spinors can be found in any book on quantum mechanics or quantum field theory. For example, see L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York 1968); A. Messiah, Quantum Mechanics (Dover, New York, 2000); L. H. Ryder, Quantum Field Theory, 2nd ed. (Cambridge U.P., Cambridge, 1996); Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory (Westview, 1995); Amitabha Lahiri and Palash B. Pal, A First Book of Quantum Field Theory (Narosa, 2000). In addition, see also R. E. Marshak, C. P. Ryan, and Riazuddin, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969), p. 82; C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw Hill, New York, 1980), p. 160; Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon, Oxford, 1984), p. 495; P. Renton, Electroweak Interactions (Cambridge U.P., Cambridge, 1990), p. 578; R. N. Mohapatra and P. B. Pal, Massive Neutrinos in Physics and Astrophysics, 2nd ed. (World Scientific, Singapore, 1998), p. 7.
11. M. Fierz, "Zur Fermischen Theorie des β-Zerfalls," Z. Phys. 104, 553 (1937).
12. E. D. Commins and P. H. Bucksbaum, Weak Interactions of Leptons and Quarks (Cambridge U.P., Cambridge, 1983).
13. The Fierz theorem is discussed in many textbooks on quantum field theory and weak interactions. See, e.g., R. E. Marshak, C. P. Ryan, and Riazuddin, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969), p. 82; C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw Hill, New York, 1980), p. 160; Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon, Oxford, 1984), p. 495; P. Renton, Electro-weak Interactions (Cambridge U.P., Cambridge, 1990), p. 578; R. N. Mohapatra and P. B. Pal, Massive Neutrinos in Physics and Astrophysics, 2nd ed. (World Scientific, Singapore, 1998), p. 7.
14. The Fierz theorem is discussed in many textbooks on quantum field theory and weak interactions. See, e.g., R. E. Marshak, C. P. Ryan, and Riazuddin, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969), p. 82; C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw Hill, New York, 1980), p. 160; Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon, Oxford, 1984), p. 495; P. Renton, Electro-weak Interactions (Cambridge U.P., Cambridge, 1990), p. 578; R. N. Mohapatra and P. B. Pal, Massive Neutrinos in Physics and Astrophysics, 2nd ed. (World Scientific, Singapore, 1998), p. 7.
15. The Fierz theorem is discussed in many textbooks on quantum field theory and weak interactions. See, e.g., R. E. Marshak, C. P. Ryan, and Riazuddin, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969), p. 82; C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw Hill, New York, 1980), p. 160; Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon, Oxford, 1984), p. 495; P. Renton, Electro-weak Interactions (Cambridge U.P., Cambridge, 1990), p. 578; R. N. Mohapatra and P. B. Pal, Massive Neutrinos in Physics and Astrophysics, 2nd ed. (World Scientific, Singapore, 1998), p. 7.
16. The Fierz theorem is discussed in many textbooks on quantum field theory and weak interactions. See, e.g., R. E. Marshak, C. P. Ryan, and Riazuddin, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969), p. 82; C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw Hill, New York, 1980), p. 160; Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon, Oxford, 1984), p. 495; P. Renton, Electro-weak Interactions (Cambridge U.P., Cambridge, 1990), p. 578; R. N. Mohapatra and P. B. Pal, Massive Neutrinos in Physics and Astrophysics, 2nd ed. (World Scientific, Singapore, 1998), p. 7.
17. The Fierz theorem is discussed in many textbooks on quantum field theory and weak interactions. See, e.g., R. E. Marshak, C. P. Ryan, and Riazuddin, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969), p. 82; C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw Hill, New York, 1980), p. 160; Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon, Oxford, 1984), p. 495; P. Renton, Electro-weak Interactions (Cambridge U.P., Cambridge, 1990), p. 578; R. N. Mohapatra and P. B. Pal, Massive Neutrinos in Physics and Astrophysics, 2nd ed. (World Scientific, Singapore, 1998), p. 7.
18. The mathematical structures of the Dirac matrices, including Fierz-like identities, have generalizations to higher-dimensional spaces [see, for example, R. Delbourgo and V. B. Prasad, "Spinor interactions in the two-dimensional limit," Nuovo Cimento Soc. Ital. Fis., A 21A, 32-44 (1974)]. In the present work we are concerned exclusively with the standard Dirac matrices appropriate to the four dimensional spacetime.
19. José F. Nieves and Palash B. Pal, "CP-violating Majorana phases, lepton-conserving processes, and final state interactions," Phys. Rev. D 67, 036005-1-036005-15 (2003).