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Paul Adrien Maurice Dirac was an English mathematical and theoretical physicist who is considered to be one of the founders of quantum mechanics. Dirac laid the foundations for both quantum electrodynamics and quantum field theory. He was the Lucasian Professor of Mathematics at the University of Cambridge, a professor of physics at Florida State University, and a 1933 Nobel Prize in Physics recipient.
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In quantum field theory, and in the significant subfields of quantum electrodynamics (QED) and quantum chromodynamics (QCD), the two-body Dirac equations (TBDE) of constraint dynamics provide a three-dimensional yet manifestly covariant reformulation of the Bethe–Salpeter equation for two spin-1/2 particles. Such a reformulation is necessary since without it, as shown by Nakanishi, the Bethe–Salpeter equation possesses negative-norm solutions arising from the presence of an essentially relativistic degree of freedom, the relative time. These "ghost" states have spoiled the naive interpretation of the Bethe–Salpeter equation as a quantum mechanical wave equation. The two-body Dirac equations of constraint dynamics rectify this flaw. The forms of these equations can not only be derived from quantum field theory they can also be derived purely in the context of Dirac's constraint dynamics and relativistic mechanics and quantum mechanics. Their structures, unlike the more familiar two-body Dirac equation of Breit, which is a single equation, are that of two simultaneous quantum relativistic wave equations. A single two-body Dirac equation similar to the Breit equation can be derived from the TBDE. Unlike the Breit equation, it is manifestly covariant and free from the types of singularities that prevent a strictly nonperturbative treatment of the Breit equation. In applications of the TBDE to QED, the two particles interact by way of four-vector potentials derived from the field theoretic electromagnetic interactions between the two particles. In applications to QCD, the two particles interact by way of four-vector potentials and Lorentz invariant scalar interactions, derived in part from the field theoretic chromomagnetic interactions between the quarks and in part by phenomenological considerations. As with the Breit equation a sixteen-component spinor Ψ is used.
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References
- Anderson, James L.; Bergmann, Peter G. (1951). "Constraints in covariant field theories". Physical Review. Series 2. 83 (5): 1018–1025. Bibcode:1951PhRv...83.1018A. doi:10.1103/PhysRev.83.1018. MR 0044382.
- Dirac, Paul A. M. (1964). Lectures on quantum mechanics. Belfer Graduate School of Science Monographs Series. Vol. 2. New York: Belfer Graduate School of Science. ISBN 9780486417134. MR 2220894. 2001 reprint by Dover.
Footnotes
- ↑ Dirac 1964, p. 43.
- ↑ Dirac 1964, p. 8.
- ↑ Dirac 1964, p. 14.
- ↑ Anderson & Bergmann 1951, p. 1019.
- ↑ Dirac, Paul A. M. (1950). "Generalized Hamiltonian dynamics". Canadian Journal of Mathematics . 2: 129–148. doi: 10.4153/CJM-1950-012-1 . ISSN 0008-414X. MR 0043724. S2CID 119748805.
- ↑ Dirac, Paul A. M. (1958). "Generalized Hamiltonian dynamics". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 246 (1246): 326–332. Bibcode:1958RSPSA.246..326D. doi:10.1098/rspa.1958.0141. ISSN 0962-8444. JSTOR 100496. MR 0094205. S2CID 122175789.
- ↑ Dirac, Paul A. M. (1958). "The theory of gravitation in Hamiltonian form". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 246 (1246): 333–343. Bibcode:1958RSPSA.246..333D. doi:10.1098/rspa.1958.0142. ISSN 0962-8444. JSTOR 100497. MR 0094206. S2CID 122053391.
- ↑ Dirac 1964.