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XV.—Quantum Mechanics of Fields. III. Electromagnetic Field and Electron Field in Interaction
Published online by Cambridge University Press: 14 February 2012
Extract
Studying the interaction of different pure fields, we have been led to some essential modifications of the ideas on which our quantum mechanics of fields is based. We shall explain these here for the example of the interaction of the Maxwell and the Dirac field.
In Part I we showed that a pure field in a given volume Ω can be described by considering the potentials and field components as matrices, not attached to single points in Ω (as the theory of Heisenberg and Pauli), but to the whole volume. Further, we assumed the total energy and momentum to be the product of Ω and the corresponding densities. In Part † we showed that this conception has to be modified; the eigenvalues of the energy and momentum as defined in Part I represent neither the states of single particles nor of a system of particles, but of something intermediate which corresponds to the simple oscillators of Heisenberg-Pauli and which we have called apeirons. The total energy and momentum of the system is a sum over the contributions of an assembly of apeirons. Mathematically the differences of the quantum mechanics of a field from that of a set of mass points (as treated in ordinary quantum mechanics) is the fact that the matrices representing a field are reducible (while those representing co-ordinates of mass points are irreducible); each irreducible submatrix corresponds to an apeiron.
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 62 , Issue 2 , 1946 , pp. 127 - 137
- Copyright
- Copyright © Royal Society of Edinburgh 1946
References
page 127 note * These Proceedings, LXII, 1944, 40.Google Scholar
page 127 note † Ibid., 92.
page 128 note † This section represents a generalisation of the introduction of the interaction with Maxwell's field given by , Pauli, Rev. Mod. Phys., XIII, 1941, 207.Google Scholar
page 131 note † Unlike the usual practice, no normalisation factor is introduced in the Fourier analysis. The Fourier coefficients of a quantity are then of the same physical dimension as that of the quantity itself.
page 133 note † , Jordan and , Wigner, Zeits.f. Physik, XLVII, 1928, 631Google Scholar.
page 133 note ‡ Novobatzky, loc. cit.
page 133 note § Since the order of factors can be arbitrarily changed in a q-number expression but not so in a q-number expression the q-number expression of (3.15) (as well as that of (3.14), etc.) is slightly ambiguous. This has no effect on the anti-commutation laws or commutation laws. The zero-point momentum and charge, however, can be avoided by taking the mean of the two possible q-number expressions as discussed in Part I.
page 135 note † Such a matrix will be referred to in what follows as a “total matrix.”