arXiv subjects
Publications and source records attributed to Nathan Poliatzky.
Recently R. G. Newton published a comment criticizing the methods and the results of a paper published by the author. His criticism touches on a few key points of the subject and hence deserves a detailed reply. Here is the reply, point by point, to his criticism.
We study a model based on $N$ scalar complex fields coupled to a scalar real field, where all fields are treated classically as c-numbers. The model describes a composite particle made up of $N$ constituents with bare mass $m_0$ interacting both with each other and with themselves via the exchange of a particle of mass $μ_0$. The stationary states of the composite particle are described by relativistic Hartree's equations. Since the self-interaction is included, the case of an elementary particle is a nontrivial special case of this model. Using an integral transform method we derive the exact ground state solution and prove its local stability. The mass of the composite particle is calculated as the total energy in the rest frame. For the case of a massless exchange particle the mass formula is given in closed form. The mass, as a function of the coupling constant, possesses a well pronounced minimum for each value of $μ_0/m_0$, while the absolute minimum occurs at $μ_0=0$.
We show that the normalization integral for the Schrödinger and Dirac scattering wave functions contains, besides the usual delta-function, a term proportional to the derivative of the phase shift. This term is of zero measure with respect to the integration over momentum variables and can be discarded in most cases. Yet it carries the full information on phase shifts and can be used for computation and manipulation of quantities which depend on phase shifts. In this paper we prove Levinson's theorem in a most general way which assumes only the completeness of states. In the case of a Dirac particle we obtain a new result valid for positive and negative energies separately. We also make a generalization of known results, for the phase shifts in the asymptotic limit of high energies, to the case of singular potentials. As an application we consider certain equations, which arise in a generalized interaction picture of quantum electrodynamics. Using the above mentioned results for the phase shifts we prove that any solution of these equations, which has a finite number of bound states, has a total charge zero. Furthermore, we show that in these equations the coupling constant is not a free parameter, but rather should be treated as an eigenvalue and hence must have a definite numerical value.
Levinson's theorem for the Dirac equation is known in the form of a sum of positive and negative energy phase shifts at zero momentum related to the total number of bound states. In this letter we prove a stronger version of Levinson's theorem valid for positive and negative energy phase shifts separately. The surprising result is, that in general the phase shifts for each sign of the energy do not give the number of bound states with the same sign of the energy (in units of $π$), but instead, are related to the number of bound states of a certain Schrödinger equation, which coincides with the Dirac equation at zero momentum.