Y. Nogami

Many-body system with a four-parameter family of point interactions in one dimension

We consider a four-parameter family of point interactions in one dimension. This family is a generalization of the usual -function potential. We examine a system consisting of many particles of equal masses that are interacting pairwise through such a generalized point interaction. We follow McGuire who obtained exact solutions for the system when the interaction is the -function potential. We find exact bound states with the four-parameter family. For the scattering problem, however, we have not been so successful. This is because, as we point out, the condition of no diffraction that is crucial in McGuires method is not satisfied except when the four-parameter family is essentially reduced to the -function potential.

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 is to 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 Appendix.

Dirac's hole theory versus quantum field theory

Diracs hole theory and quantum field theory are usually considered equivalent to each other. The equivalence, however, does not necessarily hold, as we discuss in terms of models of a certain type. We further suggest that the equivalence may fail in more general models. This problem is closely related to the validity of the Pauli principle in intermediate states of perturbation theory. PACS Nos.: 03.65-w, 11.10-z, 11.15Bt, 12.39Ba

NONLINEAR DIRAC SOLITON IN AN EXTERNAL FIELD

The behaviour of the nonlinear Dirac soliton in an external potential in 1+1 dimensions is examined by means of a collective variable ansatz and also by solving the nonlinear Dirac equation numerically. When the potential is linear with respect to coordinate x, the motion of the soliton centroid is found to be consistent with the classical relativistic equation of motion for a point particle. For a general potential there is a deviation from the behaviour of the corresponding classical point particle. This deviation can be interpreted as being caused by the finite size of the soliton.

Two definitions of the electric polarizability of a bound system in relativistic quantum theory

For the electric polarizability of a bound system in relativistic quantum theory, there are two definitions that have appeared in the literature. They differ depending on whether or not the vacuum background is included in the system. A recent confusion in this connection is clarified.

Zel'dovich's method of perturbation theory in quantum mechanics

Many years ago Zeldovich showed how the Lagrange condition in the theory of differential equations can be utilized in the perturbation theory of quantum mechanics. Zeldovichs method enables us to circumvent the summation over intermediate states. As compared with other similar methods, in particular the logarithmic perturbation expansion method, we emphasize that this relatively unknown method of Zeldovich has a remarkable advantage in dealing with excited states. That is, the ground and excited states can all be treated in the same way. The nodes of the unperturbed wavefunction do not give rise to any complication.