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Zeitschrift für Naturforschung A

Zur Interpretierbarkeit einer Quantentheorie mit indefiniter Metrik

Rolf Hüper

Published May 1, 1962• 0 citations
Publisher

Abstract

A special field theoretical model is constructed to study the problem of probability interpretation of a quantum theory with indefinite metric. The model is based on the HEISENBERG dipole-ghost version of the LEE model, but with an additional V Θ-interaction. This interaction yields bound states in the sector N+2 Θ. One gets a situation similar to a case studied by PAULI and KÄLLÉN. It seems that the arising difficulty can be solved only by means of a boundary condition suggested by BOGOLIUBOV. However, such a condition excludes the possibility of interpreting our bound states as “physically realizable” states.

DOI
10.1515/zna-1962-0503

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A special field theoretical model is constructed to study the problem of probability interpretation of a quantum theory with indefinite metric. The model is based on the HEISENBERG dipole-ghost version of the LEE model, but with an additional V Θ-interaction. This interaction yields bound states in the sector N+2 Θ. One gets a situation similar to a case studied by PAULI and KÄLLÉN. It seems that the arising difficulty can be solved only by means of a boundary condition suggested by BOGOLIUBOV. However, such a condition excludes the possibility of interpreting our bound states as “physically realizable” states.
The purpose of the following paper, which is published in three parts, is to show the possibility of making consistent use of an indefinite metric in the space of states and of the probability interpretation of quantum mechanics. In the part I bilinear forms, which are invariant under the transformations of symmetry groups, are constructed. The construction of the fundamental metric tensor for the various types of representations is exemplified for the case of inhomogenious LORENTZ group. The results are also applicable for other groups. The representations of the finite and compact group are however normal. In these cases the use of an indefinite metric does not bring about a new look for the theory of representations. In the parts II and III besides invariance of bilinear forms further conditions are stated, which are sufficient for the propability interpretation. The existence of superselection rules by which the space of states breaks up into coherent sectors leads to two procedures. In the part II the facts within one coherent sector are solely studied. These results are important for the representations of those symmetry groups, which leave the coherent sectors invariant (e. g. the inhomogenious LORENTZ group). Each sector is divided by a cut into a subspace, the elements of which represent physical systems, and a rest. The representations in the „subspace of physical systems“ of these symmetry groups, which leave the sectors invariant, are unitary. In the part III the space of states as a whole is investigated. The symmetry group of isotopic spin for a simple case is discussed there as an example of mapping from one coherent sector to another. At the end a possible generalization for the dual vector of a vector is discussed.
The content of the part I was the construction of the fundamental metric tensor; in the part II the conditions which make possible the probability interpretation for elements lying in the same coherent sector were discussed. The subject of the following part is the space of states as a whole. The existence of superselection rules allows one to give up the fundamental tensors formerly introduced as identity matrix in the subspace of the physical states. The mathematical form, in which symmetry groups appear is shown by an example. Finally a possible generalization for the dual element of an element is given.
The structure of quantum theories with indefinite metric is studied with the aid of several simple models. It is shown that the pseudo-unitary scattering matrices entering such a theory are not inconsistent with physical interpretation provided a suitable invariant projection of physical state is carried out from among all the states. A relativistic quantum theory of interacting fields is outlined and is suggested as a basis for a dynamical theory of elementary particles. It is argued that the formal introduction of an indefinite metric together with supplementary interpretive postulates may help to reinstate the principle of simplicity in a consistent theory of elementary particles.
The non-local boundary condition introduced by Bogolyubov et al. is useful for eliminating the difficulty of negative probability concerning the indefinite metric in Hilbert space. Their theory is developed in the S-matrix formalism and does not give a causal description of the physical state. A causal description is given here in the form of an ordinary Schrödinger equation. The Hamiltonian is not hermitian in general, though it gives the unitary S-matrix. Such a characteristic situation is discussed by using a simple model.
Several simple models, similar to that of Lee, involving indefinite metric are studied in this paper. In this connection, a dispersion-theoretic treatment is applied to a simple "equal-mass" model. It is shown that, at least for these models, the scattering amplitude is analytic in the upper-half energy plane provided time-reversal invariance holds; the rules of the dispersion-theoretic formulation in the case of an indefinite metric theory are given. The solution is reinterpreted as the exact solution of a slightly different model, which can also be obtained by Hamiltonian techniques; further techniques are generalized to include recoil in a relativistic no-pair model. Certain basic questions of interpretation are discussed in some detail in the concluding section.