Validity of Feynman’s prescription of disregarding the Pauli principle in intermediate states

In his space-time approach to quantum electrodynamics, Feynman advocated: “It is obviously simpler to disregard the exclusion principle completely in the intermediate states” [1]. He examined processes involving several particles and observed that all virtual processes that violate Pauli’s exclusion principle (formally) cancel out. It is understood that all virtual processes of the same orders are taken into account. On the basis of this observation Feynman introduced a prescription that is to disregard the Pauli principle in all intermediate states. This ingenious trick was crucial in accomplishing the enormous simplification and transparency of perturbation theory. For example, the vacuum polarization can be related to Feynman diagrams with an electron loop or loops. Although the process represented by a loop diagram may (at least partially) violate Pauli’s exclusion principle, no restriction needs to be imposed on integrations with respect to associated momentum variables. Feynman’s prescription is also often used in perturbation calculations for many-body systems in quantum mechanics. Various aspects of Feynman’s prescription have been discussed by several authors [2]. There are some intriguing implications regarding the meson effects in nuclei or nuclear matter. Feynman’s prescription is instrumental in proving Goldstone’s theorem for many-body systems. We are not going to review these topics in this paper but we emphasize that no suspicion seems to have ever been raised in the literature against the validity of Feynman’s prescription. The purpose of this paper we present an example that casts doubt about the general validity of Feynman’s prescription. The example is concerned with the second order energy shift of a relativistic bound system. We consider a model that consists of a particle bound in a given potential. The wave function of the particle is subject to the Dirac equation with the given binding potential. In addition to the bound particle, there is a vacuum background. It is understood that the vacuum background is an integral part of the bound system. When an external perturbation is applied, the energy of the system is shifted. We calculate the energy shift in second order perturbation theory. We are particularly interested in the vacuum effect to which the Pauli principle is relevant. We consider two methods, I and II, for calculating the energy shift. In method I we take account of the Pauli principle whenever it is applicable. In method II we disregard the Pauli principle altogether. We confirm that these two methods formally agree. This illustrates Feynman’s prescription. When methods I and II are explicitly worked out for the example, however, the results of the two methods turn out to disagree with each other. We analyze the intriguing mechanism of this discrepancy. In Sec. II we set up the model and illustrate Feynman’s prescription. In Sec. III we make the model more explicit. We consider a charged particle that is bound in an infinite square-well potential of the Lorentz scalar type. This is a one-dimensional version of the “bag model”. For the external perturbation we assume a homogeneous electric field. Then the second order energy shift is related to the electric polarizability of the system. We carry out the calculations of methods I and II. The two methods result in different energy shifts (and hence different values of the electric polarizability). We analyze the source of the discrepancy. In Sec. IV we confirm the result of method II by repeating the calculation by using the Dalgarno-Lewis (DL) method [3–5]. A summary and discussions are given in Sec. V. Some details concerning the series that appear in method II are relegated to the