A model with superconducting solution in quantum field theory

Abstract

By considering a theory based on a simple model the mathematical possibilities are analysed when the same energy operator may have in quantum theory two (approximatively) orthogonal ground states and eigenfunction systems. The consequences for the elementary particle physies are discussed.1. In theories of solid state physics based on certain models one can show that while the ground state of the unperturbed eaergy H 0 is of a "normal" type, the grouad state of the total energy H has a "superconducting" character. This latter caImot be considered as the perturbation of the normal ground state, since it is non-regular in the coupling constant. Since the two types of states are connected by a HAAG transformation [1], they ate approximatively or (when the degrees of freedom of the system grow to infinity) exactly orthogonal. The field theoretical role of sucll superconducting solutions was emphasized first by ~ 9[2]. Recently MXRSnAK and OKVBO [3] considered the possibility that the same Hamiltonian may have two types of solutions, a normal solution which may be obtained by the perturbation theory starting from the mathematical vacnum state, and a superconducting solution depending singularly on the coupling constant, the two solutions being orthogonal to each other. Thus one field operator, one Hamihonian may describe two types of particles, e. g. leptons and baryons.In the following a simple one-dimensional model having two types of solutions will be discussed and some remarks about the existence of such a twofold system of solutions in quantum field theory will be added.2. Let us considera one-dimensional anharmonic oseillator as ottr first model, having the Hamiltonian

Модель С Решением Сверхпроводимости В Квантовой Теории Поля. II.

“…It will be shown in the eourse of the caleulations that the expansion in a series in powers of the coupling eonstant 2 o ean be avoided. This expansion is, asa matter of faet, forbidden, as we have pointed out earlier [3]. We shall see that the zero point fluctuations give rise to ah iinportant interaetion ainong the field oseillators of different wave nuinber.…”

“…In the paper [3] we have made a more detailed study of the case of the asymmetric potential valley when the non-linear field equation has two different stable and constant solutions in the classical theory. In this case the symmetry (7) does not occur (because of the quadratic term).…”

“…In a previous paper [3] we eonsidered as a perturbation the terins ~a and ~4 in the expression of V(~v) which eouple oseillators of different wave number. In this treatment the interaetion induced by the zero point fluetuations of the field was omitted.…”