Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

INDEFINITE METRIC IN THE SPACE OF STATES AND PROBABILITY INTERPRETATION. PART III

Siegfried Schlieder

Published Jul 1, 1960• 0 citations
OpenAlex

Abstract

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)

Related papers

Indefinite Metrik im Zustandsraum und Wahrscheinlichkeitsinterpretation

Siegfried Schlieder

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.

Zeitschrift für Naturforschung A•Jul 1, 1960•2 citations

Indefinite Metrik im Zustandsraum und Wahrscheinlichkeitsinterpretation

Siegfried Schlieder

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.

Zeitschrift für Naturforschung A•Jun 1, 1960•7 citations

Indefinite Metrik im Zustandsraum und Wahrscheinlichkeitsinterpretation

Siegfried Schlieder

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.

Zeitschrift für Naturforschung A•Jun 1, 1960•2 citations

PROBABILITY SPACES IN QUANTUM MECHANICS. I

A. Medina

An analysis of some of the fundamental principles underlaying the construction of operators in quantum mechanics is given. Each operator is associated to a projection operator and these projection operators are shown to fulfill a set of properties formally identical to those of a probability field. It is shown that, by the introduction of a measure in configuration space it is possible to obtain complete sets of commuting observables having, within limits, preassigned spectra. The introduction of new operators by consideration of the groups expressing the symmetries of the system, is discussed. (auth)

•Jan 1, 1957•0 citations

SYMMETRIES AND QUANTIZATION: STRUCTURE OF THE STATE SPACE

P. P. Divakaran

This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.

Reviews in Mathematical Physics•Apr 1, 1994•9 citations

COHERENT STATES AND THEIR GENERALIZATIONS: A MATHEMATICAL OVERVIEW

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau +1 more

We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.

Reviews in Mathematical Physics•Oct 1, 1995•149 citations
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)
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.
An analysis of some of the fundamental principles underlaying the construction of operators in quantum mechanics is given. Each operator is associated to a projection operator and these projection operators are shown to fulfill a set of properties formally identical to those of a probability field. It is shown that, by the introduction of a measure in configuration space it is possible to obtain complete sets of commuting observables having, within limits, preassigned spectra. The introduction of new operators by consideration of the groups expressing the symmetries of the system, is discussed. (auth)
This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.
We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.