Abstract
In 30 years the science of elementary particles has made few achievements compared with its unsuccessful essays. The recent works of Schwinger, Tomonaga, Feynman and Dyson, however, have had some success (Particularly with relativistic phenomena and the relativistic subtraction). We have here a hint that progress is being made on the formal side of the discipline – though even this work is profoundly disturbing in some of its purely mathematical aspects (Cf. also Schwarz (1954). In a recent lecture at Cambridge Heisenberg remarked how the technique of ‘renormalization’, – an important formal innovation due largely to H. Bethe, – leads to the introduction of non-Hermitean operators which ruin the unitary character of the scattering matrices. This is really a fundamental change in the theory, not simply an ingenious bit of repair-work. It is objectionable because it leads to oddities like negative probabilities and experimentally vacuous ‘ghost’ states). There could be no better time to review the situation from a physical and philosophical standpoint, even if this proves to be an over-ambitious undertaking.
A revised version of an article appearing in Scientia, March, 1956.
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- 1.
For an admirable expression of this attitude, see P. E. Hodgson (1954). Hodgson remarks that “… the discovery of the nature and properties of the fundamental particles is almost entirely a matter of pure experimental research. There are no comprehensive theories which could aid the experimentalist in his investigations. Practically the only theoretical laws which can be used with a high degree of confidence are such fundamental ones as the conservation of mass-energy, charge and spin” (52). This is a little hard, but it does indicate how little elementary particle theory serves the experimenter at the present time.
- 2.
Cf. also Schiff (1949, 325).
- 3.
Removing, e.g., the degeneracy between the 2S1/2 and 2P1/2 levels.
- 4.
The electric and magnetic vectors cannot be measured in the laboratory.
- 5.
This can be derived from the Hamiltonian density referred to above, although Heisenberg’s aim was to evade the Hamiltonian formalism.
- 6.
Perhaps ‘diagrammatic innovations’ is not quite the right way of putting it. It may be argued that Feynman and Dyson have here a real physical idea, and not just a formal gadget.
- 7.
It is of course true that Feynman has constructed transition matrices in quantum electrodynamics that take into account processes of successively higher orders. E.g. one electron may interact with another by emitting a virtual photon which is absorbed by the second electron. The probability amplitude for the first electron (I) going from A to B, and the second (II) from C to D, is conceived as follows: it is the sum of the amplitude that they proceed freely (the possibility of exchange allowed for) and of the series of amplitudes corresponding (a) to interactions between the electrons and the photon fields, (b) to the creation of virtual pairs, (c) to the annihilation of created positrons with one or both of the original electrons,… and so on. The terms in this series are structured so as to exhibit the steps in each process. Thus (I) may be propagated from A to P, emit a photon, and then go on to B; while (II) is propagated from C to X, absorbs the photon (from P) and then goes on to D. The total amplitude due to this interaction is obtained by summing over all possible pairs of points P and X. Formally, this resembles the calculation of probabilities in the statistics of a stochastic process. But the quantity represented is not itself a probability – it is a probability amplitude. So that although the processes can be thought to occur as per the formalism, the propagation of probability does not follow classical statistics (because the interference of wave-mechanical probabilities plays an essential role – and operators for the creation and annihilation of particles are represented by complex numbers).
- 8.
The further you move from the ‘equator’ on a standard Mercator global projection the more uncertain you become as to the true character of the area being mapped. This distortion is a feature of the projection, not of the Polar Regions themselves. This is, in effect, the argument of D. Bohm (1952). He reconsiders the possibility of there being ‘hidden parameters’ in quantum physical phenomena, our ignorance of which requires the conceptual limitations set out in the uncertainty relations. For the purposes of this article, the author also espouses this attitude. In other contexts however, he would wish seriously to quarrel with Bohm’s thesis, as well as that of De Broglie, Einstein, Podolsky, Rosen and Jeffreys.
- 9.
There are, of course, physicists who would quarrel with this statement, Schrödinger and March, for example. But the strict continuity of these ‘classical’ versions of wave mechanics are quite useless for many-body problems, as is well-known. Thus, for any N interacting particles, the theoretician requires a 3N dimensional phase space for his calculations. The three-body problem must be worked out in a nine dimensional configuration space, which (as a literal account of a physical situation) is unintelligible to the experimenter. So the wave functions must be interpreted statistically, i.e. as giving the density of electrons within some ‘classical’ volume element, or the probability of finding a particular particle within such a volume element. So we agree with Born that the particle is the physical reality, as it would in any event be natural to suppose from the most cursory consideration of collision behaviour.
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Hanson, N.R. (2020). On Elementary Particle Theory. In: Lund, M.D. (eds) What I Do Not Believe, and Other Essays. Synthese Library, vol 38. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1739-5_2
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