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Born's electrodynamics in complex form

Published online by Cambridge University Press:  24 October 2008

P. Weiss
P. Weiss
Affiliation:
Downing College
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Extract

Born's electrodynamics, together with its quantization, has been developed in several papers by Born and Infeld†. It is a general concept, containing many different systems of electrodynamics, of which Maxwell's theory is one particular case. Each system is characterized by a Lagrangian (or Hamiltonian) function of two invariants F and G (or P and Q). These invariants are built up by means of two antisymmetrical tensors, which are the “field quantities”.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 33 , Issue 1 , March 1937 , pp. 79 - 93
Copyright
Copyright © Cambridge Philosophical Society 1937

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References

† Born, M., Proc. Roy. Soc. A, 143 (1934), 410,CrossRefGoogle Scholar quoted as I; Born, M. and Infeld, L., Proc. Roy. Soc. A, 144 (1934), 425,CrossRefGoogle Scholar quoted as II; Born, M. and Infeld, L., Proc. Roy. Soc. A, 147 (1934), 522,CrossRefGoogle Scholar quoted as III; Born, M. and Infeld, L., Proc. Roy. Soc. A, 150 (1935), 141,Google Scholar quoted as IV; Infeld, L., Proc. Camb. Phil. Soc. 32 (1936), 127,CrossRefGoogle Scholar quoted as V.

‡ Schrödinger, E., Proc. Roy. Soc. A, 150 (1935), 465.CrossRefGoogle Scholar

† II, pp. 431, 435. Our F and P differ from the previous ones by a factor ½. This turns out to be more convenient, because it avoids unnecessary numerical factors.

‡ The second equation of (1·5) and the first of (1·7) are given in II as (2·25) and (4·17).

Note added in proof. Equations (1·2)–(1·4a) have recently been given by Madhava Rao, B. S., cf. Proc. Ind. Acad. Science, iv, 5 (1936), pp. 578–9,Google Scholar and applied to the “new field theory”.

† Cf. Weyl, H., Raum, Zeit, Materie, 5th ed. (Berlin, 1923), pp. 232–5.CrossRefGoogle Scholar

‡ Compare IV p. 159.

† Cf. II, pp. 434, 436.

† The difference in the sign could have been avoided if, in order to make the symmetry between f kl and complete, we had written instead of p kl.

‡ This T differs from the one defined by Infeld by the factor ½. Compare V p. 129.

† Cf. Klein, F., Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, 2 (Berlin, 1927), p. 82.Google Scholar

‡ This transformation has a close connection with the formalism of spinors and semivectors. Cf. e.g. Einstein, A. and Mayer, W., Berl. Sitz. Ber. (1932), pp. 526–6;Google ScholarMadhava Rao, B. S., Proc. Ind. Acad. Science iv, 4 (1936), pp. 437–9;Google ScholarWhittaker, E. T., Proc. Roy. Soc. 158 (1937), pp. 38–41.CrossRefGoogle Scholar

† Compare V p. 130 (3·6). There the imaginary parts of ø, ψ, ρ are zero.

† Cf. V p. 129.

‡ This fact, though it has never been stated explicitly, arises from the method of formation of the invariants. Cf. II, pp. 430–1.

§ Let ω (z) = u (x, y) + iv (x, y) be such that u (x, y) = u (x, − y). Then, if

the Cauchy-Riemann equations show that

neglecting a trivial constant. Hence

∥ An example of a many-valued function will be discussed in § 8.

† This condition is already sufficient for self-conjugacy. Cf. Titchmarsh, , Theory of functions (Oxford, 1932), p. 155.Google Scholar

‡ The weaker property that assumes conjugate complex values for conjugate complex values of the tensor πkl (or ψkl) follows also from the equations (6·9) and (6·16). Our proposition is much stronger and its validity is due to the more far-reaching relation (6·19). On the other hand it will be seen from (6·9) that ψkl does not assume conjugate complex values for conjugate complex values of økl, but that

and vice versa.

† I, II. The absolute field is taken as unity.

† Compare I p. 427.

‡ V, pp. 129–30, 134–5.

§ Viz. V p. 134, formulae (7·1), (7·2).