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Zeitschrift für Naturforschung A

Indefinite Metrik im Zustandsraum und Wahrscheinlichkeitsinterpretation

Siegfried Schlieder

Published Jun 1, 1960• 7 citations

Abstract

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.

DOI: 10.1515/zna-1960-5-610

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Zur Interpretierbarkeit einer Quantentheorie mit indefiniter Metrik

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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.

Zeitschrift für Naturforschung A•May 1, 1962•0 citations

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 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)

We present the theoretical considerations for the case of looking into a generalization of quantum theory corresponding to having an inner product with an indefinite signature on the Hilbert space. The latter is essentially a direct analog of having the Minkowski spacetime with an indefinite signature generalizing the metric geometry of the Newtonian space. In fact, the explicit physics setting we have in mind is exactly a Lorentz covariant formulation of quantum mechanics, which has been discussed in the literature for over half a century yet without a nice full picture. From the point of view of the Lorentz symmetry, indefiniteness of the norm for a Minkowski vector may be the exact correspondence of the indefiniteness of the norm for a quantum state vector on the relevant Hilbert space. That, of course, poses a challenge to the usual requirement of unitarity. The related issues will be addressed.

Noninertial transformations on time–position–momentum–energy space { t , q , p , e } with invariant Born–Green metric and the symplectic metric −d e ∧ d t + d p ∧ d q are studied. This group of transformations contains the Lorentz group as the inertial special case and, in the limit of small forces and velocities, reduces to the expected Hamilton transformations leaving invariant the symplectic metric and the nonrelativistic line element d s 2 = −d t 2 . The transformations bound relative velocities by c and relative forces by b . Spacetime is no longer an invariant subspace but is relative to noninertial observer frames. In the limit of b → ∞, spacetime is invariant. Born was lead to the metric by a concept of reciprocity between position and momentum degrees of freedom and for this reason we call this reciprocal relativity. For large b , such effects will almost certainly only manifest in a quantum regime. Wigner showed that special relativistic quantum mechanics follows from the projective representations of the inhomogeneous Lorentz group. Projective representations of a Lie group are equivalent to the unitary representations of its central extension. The same method of projective representations for the inhomogeneous group is used to define the quantum theory in the noninertial case. The central extension of the inhomogeneous group is the cover of the quaplectic group . is the Weyl–Heisenberg group. The group, and the associated Heisenberg commutation relations central to quantum mechanics, results directly from requiring projective representations. A set of second-order wave equations result from the representations of the Casimir operators.

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.