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The calculations of processes involving mesons by matrix methods

Published online by Cambridge University Press:  24 October 2008

A. H. Wilson
A. H. Wilson
Affiliation:
Trinity CollegeCambridge
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Extract

In a previous paper a new method, based on Kemmer's β-formalism, of calculating meson processes was given for the case in which the meson interacts with an electromagnetic field. This method is now extended to the nuclear interaction, so that the whole of the meson theory can be given either in tensor or in matrix form, the former being preferable when the wave aspect of the meson is important and the latter when the particle aspect is dominant.

As examples of the matrix method, derivations are given of the cross-sections for the nuclear scattering of mesons and for the production of mesons from nuclei by photons. It is pointed out that the usual non-relativistic theory of the nuclear interaction is inadequate even for very small velocities.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 36 , Issue 3 , July 1940 , pp. 363 - 380
Copyright
Copyright © Cambridge Philosophical Society 1940

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References

§ Proc. Roy. Soc. A, 173 (1939), 91.Google Scholar An equivalent formulation which includes the nuclear interaction has been given in spinor notation by Belinfante, , Dissertation (Leiden, 1939)Google Scholar, but this is not so convenient as Kemmer's for our purpose.

∥ Wilson, and Booth, , P.roc. Roy. Soc.Google Scholar A (in the Press), referred to as I.

§ Proc. Roy. Soc. A, 166 (1938), 501.Google Scholar

§ No confusion between the 4 × 4 matrix β and the 10 × 10 matrices βμ will arise if it is kept in mind that the 10 × 10 matrices either have a suffix attached or appear as components of a vector β printed in Clarendon type.

∥ In order to have constants with convenient dimensions we alter Bhabha's notation at this point. In his notation, our g 1 and g 2 are replaced by and With our choice of the constants, the potential energy, due to the g 1 term, of a proton in the presence of a neutron is .

It may be remarked that the present formulation can be easily extended so as to include all the cases of particles with spin 0 or 1 considered by Kemmer, (Proc. Roy. Soc. A, 166 (1938), 127)CrossRefGoogle Scholar, the only essential difference being in the explicit form of ξ. In the pseudo-vector theory ξ is a ten-component quantity as here, while in the scalar and pseudo-scalar theories ξ and ψ are five-component quantities and βμ is a 5 × 5 matrix.

Note that in (10) there are three distinct types of matrices to be multiplied. The matrices τ1 and τ2 operate on Ψ, the two components ΨP and ΨN of Ψ being themselves four-component matrices. The ten-component matrices ψ† and ξ, and ξ† and ψ, are multiplied together, the components of ξ and ξ† being 4 × 4 matrices which are multiplied on the left by Ψ†P and Ψ†N, and on the right by ΨP and ΨN. It is therefore possible to change the order of factors of different types without altering the expression.

§ See, for example, Heitler, , Quantum theory of radiation (Oxford, 1936), p. 149.Google Scholar

§ Proc. Cambridge Phil. Soc. 35 (1939), 463.Google Scholar

∥ Z. Phys. 70 (1931), 686.Google Scholar

§ The matrix elements of the interaction with respect to the proton wave function can be found for example in Heitler's book, op. cit. pp. 95 ff. Owing to a difference in normalization the matrix elements given there must be multiplied by a factor .

§ Cf. Heitler, , Proc. Roy. Soc. A, 166 (1938), 529.CrossRefGoogle Scholar

§ Kobayasi, and Okayma, , Proc. Phys.-Math. Soc. Japan, 21 (1939), 1Google Scholar. Chang, , Proc. Cambridge Phil. Soc. 36 (1940), 34.CrossRefGoogle Scholar