Summary
We show that, in addition to the usual covariant tensor form of Maxwell’s equations, a more general, manifestly covariant, Dirac bispinor form of them exists. A study of the Lagrangian formulation of electrodynamics, utilizing this more general bispinor electromagnetic-field structure, is conducted. We find that, in addition to the conventional Lorentz force interaction, a new electromagnetic « spinor-torque » interaction, related to the bispinor structure, is generated. We study the effect that Dirac bispinor electrodynamics has on the spectrum of relativistic hydrogen and show that the new « spinor-torque » interaction will contribute an extra anomalous hyperfine structure term, in addition to that generated by proton spin structure and quantum electrodynamic effects. Estimates of the value of the coupling constant associated with the « spinor-torque » interaction, which has the dimensions of a length, are given.
Riassunto
Si dimostra che, oltre alla consueta forma tensoriale covariante delle equazioni di Maxwell, ne esiste un’altra forma più generale ed evidentemente covariante del tipo bispinoriale di Dirac. Si studia la formulazione lagrangiana dell’elettrodinamica, utiliz-zando questa struttura bispinoriale più generale del campo elettromagnetico. Si trova che, oltre all’interazione convenzionale della forza di Lorentz, si genera una nuova interazione elettromagnetica « spinore-torsione », relativa alla struttura bispinoriale. Si studia l’effetto dell’elettrodinamica bispinoriale di Dirac sullo spettro dell’idrogeno relativistico e si dimostra che la nuova interazione spinore-torsione genera un termine anomalo di struttura iperfine oltre a quelli determinati dalla struttura di spin del protone e dagli effetti elettrodinamici quantistici. Si danno valutazioni del valore della costante di accoppiamento associata col nuovo tipo di interazione, che ha le dimensioni di una lunghezza.
Реэюме
Мы покаэываем, что помимо обычной ковариантной тенэорной формы уравнений Максвелла, сушествует более обшая, явно ковариантная, дираковская биспинорная форма зтих уравнений. Проводится исследование лагранжи-анной формулировки злектродинамики, испольэуя зту более обшую структуру биспи-норного злектромагнитного поля. Мы находим, что помимо обычного вэаимодействия, описываемого силой Лорентда, воэникает новое злектромагнитное вэаимодействие «спинор-врашаюший момент», свяэанное с биспинорной структурой. Мы исследуем влияние, которое окаэывает дираковская биспинорная злектродинамика на спектр релятивистского водорода, и покаэываем, что новое вэаимодействие «спинор-врашаюший момент» будет давать дополнительный аномальный член в сверхтонкую структуру, в дополнение к члену, обраэованному спиновой структурой протона и зффектами квантовой злектродинамики. Приводятся оценки величины постоянной свяэи, свяэанной с вэаимодействием «спинор-врашаюший момент», которое имеет раэмерность длины.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A clear and detailed discussion of the relativistic covariance properties of the Dirac bispinor equations is given in,S. Schweber:Introduction to Relativistic Quantum Field Theory (New York, 1962).
Use the data contained inH. Bethe andE. Salpeter:Quantum Mechanics of One and Two Electron Atoms, inHandbuch der Physik, vol.35 (Berlin, 1957), p. 149, 199.
As to the question of how the use of an additional Dirac bispinor freedom, in the electromagnetic field, affects the spin of the photon we can obtain an answer from an application of Noether’s theorem to the Lagrangian density (16) (E. Noether:Nachr. d. Kgl. Ges. d. Wiss., 235, (1918)). The spin contribution to the angular momentum of the system, associated with the electromagnetic field, isS=εdx 3[(E×A)+λ{ie61-1}Φ † Σ Φ]. The first term can be shown to yield the usual spin-one photon contribution (seeN. N. Bogoliubov:Introduction to Quantum Field Theory (New York, 1959). The second term, proportional toλ, yields the anomalous spin contribution to the photon coming from the bispinor electromagnetic field degrees of freedom. This means that in this theory the photon has an anomalous spin contribution, in addition to its usual spin-one value, proportional toλ∼10−19. Because of the extreme smallness ofλ in this theory, the effects of this anomalous photon spin would be far below the present experimental test levels of conventional quantum electrodynamics. This is essentially becauseλ measures the ratio of the physical effects of the spinor torque interaction to that of the Lorentz force interaction. The study of spinor forms of Maxwell’s equations is not a new subject. For example, seeP. Roman:Theory of Elementary Particles, Chap. 11 (Amsterdam, New York, 1960);M. Sachs andS. Schwebel:Journ. Math. Phys.,3, 843 (1962).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Leiter, D. The Dirac bispinor form of Maxwell’s equations and the anomalous hyperfine structure of relativistic hydrogen. Nuovo Cimento B (1965-1970) 68, 53–63 (1970). https://doi.org/10.1007/BF02710358
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02710358