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The wave equation in conformal space

Published online by Cambridge University Press:  24 October 2008

H. J. Bhabha
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In a recent paper Dirac has shown that by passing from the ordinary Euclidean space to a four-dimensional conformal space, some of the equations of physics can be written in a tensor form, the indices of which take on six values. Those equations which can be written in this form are then invariant under conformal transformations of the Euclidean space. Among the equations of physics which have this more general invariance are the Maxwell equations, as was proved by a direct transformation a long time ago by Cunningham, and Bateman, so that Dirac's paper provides an alternative and more general proof of this result. Certain errors ∥ in Dirac's paper, however, necessitate a reformulation of the proof. Before we do this in § 2, we briefly recapitulate in § 1 some of the general results derived there. In § 3 we investigate further the conformal invariance of the wave equation for an electron in the presence of a general electromagnetic field.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 32 , Issue 4 , December 1936 , pp. 622 - 631
Copyright
Copyright © Cambridge Philosophical Society 1936

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References

† Dirac, P. A. M., Ann. of Math. 37 (1936), 429.CrossRefGoogle Scholar

‡ Cunningham, E., Proc. Lond. Math. Soc. (2), 8 (1909), 77.Google Scholar

§ Bateman, H., Proc. Lond. Math. Soc. (2), 8 (1910), 223.CrossRefGoogle Scholar

∥ See footnote, p. 624 below.

Greek indices take on all integral values from one to six, Latin indices from one to four. We always sum over repeated suffices.

† A conformal vector A μ transforms by definition like x μ, i.e. its transformation law is the same as (2). Tensors of higher rank transform correspondingly.

‡ Dirac, loc. cit.

† The equations (11) are necessary for the progress of our argument, as well as of Dirac's. Dirae does not assume the equations (11) explicitly, but proceeds to derive them from the two equations

which he assumes to hold on the hyperquadric. An error has however occurred in his proof, and it may be shown that it is necessary to assume the conditions (11) explicitly.

† S may even be taken of the form x μx μS′, so that the equations can be brought to the form (20) without changing the values of the potentials in Euclidean space.

‡ The first of equations (20) only has a meaning on the hyperquadric, since j ν is only defined on the hyperquadric. We may, if we choose, use (17) to define j ν in the rest of space. In this case the first of the equations (20) automatically holds in all space, from (18), provided that the second of equations (20) holds in the whole of space. It should be noticed that the right-hand side of (22) only satisfies (21) on the hyperquadric, so that such a transformation leaves the equations (20) unchanged only on the hyperquadric. The divergence of the A μ does not vanish in general off the hyperquadric. However, by adding terms of the type B μx νx ν to the A μ, which leave the potentials in Euclidean space unaltered, we can make the second of the equations (20) hold in the whole of space. To do this we choose x μB μ to be of the form Bx νx ν, with a suitable choice of B and the B μ.