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Zeitschrift für Naturforschung A

Indefinite Metrik im Zustandsraum und Wahrscheinlichkeitsinterpretation

Siegfried Schlieder

Published Jun 1, 1960• 2 citations
Publisher

Abstract

In part I invariant bilinear forms were constructed by means of suitable fundamental tensors. Invariance is one of the conditions for the probability interpretation. In the following part further conditions are specified for the “space of physical states”. They are sufficient at the first instance for the probability interpretation for elements of one coherent sector.

DOI
10.1515/zna-1960-5-611

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Indefinite Metrik im Zustandsraum und Wahrscheinlichkeitsinterpretation

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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.

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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.

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In part I invariant bilinear forms were constructed by means of suitable fundamental tensors. Invariance is one of the conditions for the probability interpretation. In the following part further conditions are specified for the “space of physical states”. They are sufficient at the first instance for the probability interpretation for elements of one coherent sector.
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 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 Part I was the construction of the fundamental metric tensor; in Part II the conditions which make possible the probability interpretation for elements lying in the same coherent sector were discussed. The subject of this part (III) is the space of states as a whole. The existence of superselection rules allows the fundamental tensors formerly introduced as identity matrix in the subspace of the physical states to be given up. 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. (auth)
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.