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XII.—Quantum Mechanics of Fields. II. Statistics of Pure Fields
Published online by Cambridge University Press: 14 February 2012
Extract
The quantum mechanics of fields recently developed by us leads to a modification of statistical mechanics of elementary particles which seems to overcome some of the difficulties (divergence of integrals) occurring in the usual quantum theory of fields. The main difference between the new theory and the usual one is as follows.
In the usual theory the wave-vector k is introduced classically and, so to speak, kinematically by the Fourier analysis of the field. The Fourier coefficients of the field components are then treated according to quantum mechanics as non-commuting quantities; those belonging to the wave-vector k describe the corresponding “model” mechanical system, namely the kth radiation oscillator. But the statement that the Fourier coefficients belonging to a certain k all vanish, which statement classically is significant, is now meaningless because there is a lowest state with zero-point energy for each radiation oscillator. The field is thus made to be equivalent to the assembly of radiation oscillators of all possible wave-vectors which, being necessarily infinite in number, contribute an infinite zero-point energy for the pure field and lead to other divergent integrals for the interaction between different fields.
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 62 , Issue 1 , 1944 , pp. 92 - 102
- Copyright
- Copyright © Royal Society of Edinburgh 1944
References
page 92 note * We shall refer to Part I of this series of papers by simply quoting I. See these Proceedings, LXII, 1944, 40–57Google Scholar.
page 93 note * For the demonstration of this theorem see , Born and , Jordan, Elementare Quantenmeckanik, Springer, 1930, § 23,Google Scholar for a single particle, and § 17 for a system of particles.
page 93 note † The notion of “pure state” was used in I, section 3, in a way which did not take account of the fundamental difference between ordinary quantum mechanics and the present field theory concerning the irreducibility of the matrices. We have to correct these statements according to our present better understanding of the mathematical structure of the new theory.
page 94 note * The term ἄπειρον was introduced by Anaximander of Miletos (about 550 B.C.) for the boundless and shapeless primordial matter which is the first product (arché, ἀρχή) of the creation and develops into the specific types of ordinary matter.
page 94 note † The term radiation oscillator used throughout this paper differs from its current use in the minor point that we do not analyse it into simple harmonic oscillators, one for each polarisation.
page 95 note * It can be seen that the application of statistical mechanics to the apeirons of a pure electron (Dirac) field leads to no canonical equilibrium.
page 95 note † Here and in what follows k denotes the wave-vector of one apeiron. We regret that the same symbol has been used in I both for this and for the configuration wave-vector of the assembly of apeirons, but the context and the other symbols which occur in an equation will determine the meaning of k.
page 98 note * This follows from the adiabatic principle of quantum mechanics.
page 93 note † , Bergmann, Phys. Rev., LIX, 1941, 928.Google Scholar
page 102 note * See, for instance, , Fowler, Statistical Mechanics, Cambridge University Press, 1936Google Scholar.