It is shown how the classical Yang‐Mills (and the Maxwell) field equations can be written in a simple form by introducing differential operators based on the split‐Cayley algebra. As a result, a field algebra accommodating both space‐time symmetry (Lorentz invariance) and internal symmetry evolves. The connection between the view of the Yang‐Mills field as a split‐Cayley algebra and the exceptional Jordan angebra , discovered by Jordon, von Neumann, and Wigner, is discussed.
REFERENCES
1.
2.
P.
Jordan
, J.
von Neumann
, and E.
Wigner
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, 29
(1934
).3.
4.
5.
A. Gamba, “Proceedings of the Seminar on High Energy Physics and Elementary Particles,” Trieste, published by the International Atomic Energy Agency, 1965.
6.
In all that follows, Greek indices run over (1,2,3,4), lower case Latin indices run over (1,2,3), and upper case Latin indices refer to the internal symmetry space which is usually left unspecified. We take the Minkowski metric Tneusual summation convention is assumed. The and of Ref. 1 are related to quantities defined here by and
7.
8.
See W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison‐Wesley, Reading, Mass., 1955).
9.
I am grateful to Professor H. Brown and Professor R. Brown of The Ohio State University for valuable discussions of this paper.
10.
As long as there is one element x such that then this element can be “normalized” to give a unit.
11.
The vector potential used here differs by exactly a sign from that defined in many texts, e.g., Ref. 8. This sign difference has no effect on the equations for the electric and magnetic field intensities; our choice allows the electric field intensity to be expressed as the spatial part of the 4‐curl of A.
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© 1973 The American Institute of Physics.
1973
The American Institute of Physics
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