A note on podolsky electrodynamics

Abstract

Three formalisms have been considered for the ‘extraordinary’ field introduced by Podolsky in his generalized electrodynamics.The c-representation (tildons) for negative energy particles (§ 3), leads to an unstable vacuum in electron hole theory and must be discarded (§ 7).The d-representation suggested by Podolsky is a hole theory for bosons. This leads at once to inconsistencies and it is impossible to introduce a Fock representation (§ 6).The least unsatisfactory is the b-representation (tilde photons) used by PKMG. Tilde photons can be created at high energies but decay immediately by pair creation. The introduction of tilde photons does not interfere with any agreement already established between experiment and ordinary quantum electrodynamics. Evidence might be obtained for the value of the constant a from consideration of the 2S level shift of the hydrogen atom and from the theory of showers where the deviation from the usual theory is greatest (§ 7).However, the formalism involves the introduction of an indefinite metric for the tilde photon field and consequently ‘negative probabilities’ for states in which an odd number of tilde photons are present. Approximate results can be obtained by assuming that ‘negative probabilities’ in the theory are positive probabilities in the real world. This leads to a small but fundamental error and it does not seem possible that a theory in this form can have any deep seated validity (§ 5).The same analysis applies to the extension of the Podolsky theory to the Proca field by Green (5). In order that the energy should be positive definite it is necessary to introduce an indefinite metric for one of the two superposed meson fields. Fundamentally the difficulties of interpretation are the same but in practice they will be more obvious because the coupling constants are greater.