A graphical representation of the Dirac algebra

American Journal of Physics

Volumn 64, Issue 7, 1996, Pages 870-880

  Goodmanson, David M ^a  

^a NONE   (United States)

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[No Author keywords available]

Indexed keywords

EID: 0030537547     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18113     Document Type: Article

References (17)

1. We use natural units, ℏ = c = 1. In the "ict" convention of special relativity, an explicit factor of i multiplies the time, and usually the time coordinate is given the index 4; in covariant notation there is no factor of i and the time coordinate is given the index 0. In this paper the time coordinate always has index 0.
2. Unless otherwise indicated, indices i, j, and k run from 1 to 3 and greek indices μ and ν run from 0 to 3. Indices m through s have a larger specified domain, such as 0 to 3 and also 5. Summation convention is assumed.
3. 2 = abI. Since the diagonal elements of mm† are real and positive definite, it follows that a = b*l\b\. We can conclude that m† = (b*l\b\)m.
4. J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, Reading, MA, 1967), p. 81.
5. In a general Clifford algebra each of the gammas can have square ±I. In the original Dirac form all squares are chosen to be +I.
6. Hestenes, in D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus (Kluwer, Hingham, MA, 1984), pp. xiii, 180-182, has emphasized that Clifford algebras with complex coefficients are not really necessary, since there are equivalent Clifford algebra formulations with real coefficients. We use the first approach, with explicit factors of i, so as to closely parallel the standard representation of the Dirac algebra by gamma matrices.
7. +a is not intended here. The + convention is reserved for situations where the labels 0...,3 and 5 are explicitly involved.
8. A new choice of mapping will interchange the gamma matrices among themselves, which strictly speaking does not create a new representation but merely a permutation of the same representation. But it is standard practice in physics to call these different representations.
9. See, for example, K. Gottfried and V. F. Weisskopf, Concepts of Particle Physics (Oxford, New York, 1986), Vol. II pp. 198-204.
10. iφ =cos(φ) + i sin(φ).
11. X). Lorentz transformations are not unitary.
12. The last two operations also involve complex conjugation of the Dirac equation.
13. 0).
14. John Bell's papers on the subject are collected in J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge, New York, 1987). A good review of Bell's theorems and related topics is L. E. Ballentine, Foundations of Quantum Mechanics Since the Bell Inequalities, RB 52 (AAPT, College Park, MD, 1988).
15. John Bell's papers on the subject are collected in J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge, New York, 1987). A good review of Bell's theorems and related topics is L. E. Ballentine, Foundations of Quantum Mechanics Since the Bell Inequalities, RB 52 (AAPT, College Park, MD, 1988).
16. N. D. Mermin, "Hidden Variables and the Two Theorems of John Bell," Rev. Mod. Phys. 65, 803-815 (1993).
17. z)/√2.