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

arXiv:0809.0561 · AIP conference proceedings

On the Hermitian Projective Line as a Home for the Geometry of Quantum Theory

Wolfgang Bertram, Piotr Kielanowski, Anatol Odzijewicz, Martin Schlichenmaier, Theodore Voronov

Published Jan 1, 2008• 1 citation
arXivHTMLPDF

Abstract

In the paper [1], generalized projective geometries have been proposed as a framework for a geometric formulation of Quantum Theory. In the present note, we refine this proposition by discussing further structural features of Quantum Theory: the link with associative involutive algebras A and with Jordan‐Lie and Lie‐Jordan algebas. The associated geometries are (Hermitian) projective lines over A; their axiomatic definition and theory will be given in subsequent work with M. Kinyon [2].

DOI
10.1063/1.3043855

Related papers

Geometrical description of algebraic structures: Applications to Quantum Mechanics

José F. Cariñena, Alberto Ibort, G. Marmo +4 more

Geometrization of physical theories have always played an important role in their analysis and development. In this contribution we discuss various aspects concerning the geometrization of physical theories: from classical mechanics to quantum mechanics. We will concentrate our attention into quantum theories and we will show how to use in a systematic way the transition from algebraic to geometrical structures to explore their geometry, mainly its Jordan‐Lie structure.

arXiv:1209.4583•AIP conference proceedings•Jan 1, 2009•6 citations

All Quantum Probability viewed in Complex Projective Geometry

Stephen Bruce Sontz

In a recent paper it was shown that all the Hilbert space formulas for quantum probabilities can be realized as functions of geometric properties of the associated projective space, but those functions were expressed using the structures of the associated Hilbert space. In this paper a direct description of all these probabilities is given as formulas involving only the geometric properties of the projective space itself without referring to the associated Hilbert space theory. In large part this depends on a projection theorem for complex projective space which is analogous to the projection theorem for Hilbert spaces. The importance of this is that this exhibits quantum probability in terms of the geometry of a Riemannian metric in a non-linear Kähler manifold without any reference to a linear Hilbert space. As such this is a part of a larger program of the geometrization of physics. This opens the possibility of generalizations of quantum theory in other similar geometric settings. The theory presented includes projective spaces of both finite and infinite dimension. Some comments explain how quantum theory based on a von Neumann algebra is compatible with this approach.

quant-pharXiv:2605.17578•May 19, 2026•0 citations

An Essay on the Completion of Quantum Theory. I: General Setting

Wolfgang Bertram

We propose a geometric setting of the axiomatic mathematical formalism of quantum theory. Guided by the idea that understanding the mathematical structures of these axioms is of similar importance as was historically the process of understanding the axioms of geometry, we complete the spaces of observables and of states in a similar way as in classical geometry linear or affine spaces are completed by projective spaces. In this sense, our theory can be considered as a "completion of usual linear quantum theory" , such that the usual theory appears as the special case where a reference frame is fixed once and for all. In the present first part, this general setting is explained. Dynamics (time evolution) will be discussed in subsequent work.

arXiv:1711.08643•Nov 23, 2017•2 citations

Symmetry in Quantum Theory

Gérard G. Emch, C. Piron

The aim of this note is to give a precise definition of symmetry in quantum theory in order to generalize Wigner's representation theorem, in the framework of lattice theory and projective geometry. We do not require the concept of physical state; the results are valid for any field (division ring) used in the realization of the lattice of propositions. Physical systems with the most general superselection rules are included in the theory.

Journal of Mathematical Physics•Apr 1, 1963•43 citations

General quantum theory—unification of classical and modal quantum theories

Koen Thas

Abstract Inspired by classical (‘actual’) quantum theory over C and modal quantum theory (MQT), which is a model of quantum theory over certain finite fields, we introduce general quantum theory as a quantum theory—in the København interpretation—over general division rings with involution, in which the inner product ‘is’ a ( σ , 1)-Hermitian form φ . This unites all known such approaches in one and the same theory, and we show that many of the known results such as no-cloning, no-deleting, quantum teleportation and super-dense quantum coding, which are known in classical quantum theory over C and in some MQTs, hold for any general quantum theory. Our approach is totally different (and more general) than Hardy’s axiomatic approach [15], and generalized probability theories. On the other hand, in many general quantum theories, a geometrical object which we call ‘quantum kernel’ arises, which is invariant under the unitary group U ( V , φ ), and which carries the geometry of a so-called polar space. This object cannot be seen in classical quantum theory over C , but it is present, for instance, in all known MQTs. We use this object to construct new quantum (teleportation) coding schemes, which mix quantum theory with the geometry of the quantum kernel (and the action of the unitary group). We also show that in characteristic 0, every general quantum theory over an algebraically closed field behaves like classical quantum theory over C at many levels, and that all such theories share one model, which we pin down as the ‘minimal model,’ which is countable and defined over Q ¯ . Moreover, to make the analogy with classical quantum theory even more striking, we show that Born’s rule holds in any such theory. So all such theories are not modal at all. Finally, we obtain an extension theory for general quantum theories in characteristic 0 which allows one to extend any such theory over algebraically closed fields (such as classical complex quantum theory) to larger theories in which a quantum kernel is present. In this sense, these singular objects are always virtually around in abundance.

Journal of Physics A Mathematical and Theoretical•Jul 6, 2020•1 citation

A New Definition of Quantum Superposition

Stephen Bruce Sontz

The usual description of the superposition of two (pure quantum) states is ambiguous, since the binary operation of addition in a Hilbert space does not pass down to the quotient projective space. In the setting of complex projective space this paper gives a new geometrical definition of the superposition of two pure states, viewed as two distinct points in the projective space, as the unique (complex) line on which those two points lie. Using the Erlangen program the geometry appropriate for quantum theory of the projective space is precisely defined. Also this rigorous approach is compared with others.

quant-pharXiv:2606.15607•Jun 16, 2026•0 citations
In the paper [1], generalized projective geometries have been proposed as a framework for a geometric formulation of Quantum Theory. In the present note, we refine this proposition by discussing further structural features of Quantum Theory: the link with associative involutive algebras A and with Jordan‐Lie and Lie‐Jordan algebas. The associated geometries are (Hermitian) projective lines over A; their axiomatic definition and theory will be given in subsequent work with M. Kinyon [2].
In a recent paper it was shown that all the Hilbert space formulas for quantum probabilities can be realized as functions of geometric properties of the associated projective space, but those functions were expressed using the structures of the associated Hilbert space. In this paper a direct description of all these probabilities is given as formulas involving only the geometric properties of the projective space itself without referring to the associated Hilbert space theory. In large part this depends on a projection theorem for complex projective space which is analogous to the projection theorem for Hilbert spaces. The importance of this is that this exhibits quantum probability in terms of the geometry of a Riemannian metric in a non-linear Kähler manifold without any reference to a linear Hilbert space. As such this is a part of a larger program of the geometrization of physics. This opens the possibility of generalizations of quantum theory in other similar geometric settings. The theory presented includes projective spaces of both finite and infinite dimension. Some comments explain how quantum theory based on a von Neumann algebra is compatible with this approach.
We propose a geometric setting of the axiomatic mathematical formalism of quantum theory. Guided by the idea that understanding the mathematical structures of these axioms is of similar importance as was historically the process of understanding the axioms of geometry, we complete the spaces of observables and of states in a similar way as in classical geometry linear or affine spaces are completed by projective spaces. In this sense, our theory can be considered as a "completion of usual linear quantum theory" , such that the usual theory appears as the special case where a reference frame is fixed once and for all. In the present first part, this general setting is explained. Dynamics (time evolution) will be discussed in subsequent work.
Abstract Inspired by classical (‘actual’) quantum theory over C and modal quantum theory (MQT), which is a model of quantum theory over certain finite fields, we introduce general quantum theory as a quantum theory—in the København interpretation—over general division rings with involution, in which the inner product ‘is’ a ( σ , 1)-Hermitian form φ . This unites all known such approaches in one and the same theory, and we show that many of the known results such as no-cloning, no-deleting, quantum teleportation and super-dense quantum coding, which are known in classical quantum theory over C and in some MQTs, hold for any general quantum theory. Our approach is totally different (and more general) than Hardy’s axiomatic approach [15], and generalized probability theories. On the other hand, in many general quantum theories, a geometrical object which we call ‘quantum kernel’ arises, which is invariant under the unitary group U ( V , φ ), and which carries the geometry of a so-called polar space. This object cannot be seen in classical quantum theory over C , but it is present, for instance, in all known MQTs. We use this object to construct new quantum (teleportation) coding schemes, which mix quantum theory with the geometry of the quantum kernel (and the action of the unitary group). We also show that in characteristic 0, every general quantum theory over an algebraically closed field behaves like classical quantum theory over C at many levels, and that all such theories share one model, which we pin down as the ‘minimal model,’ which is countable and defined over Q ¯ . Moreover, to make the analogy with classical quantum theory even more striking, we show that Born’s rule holds in any such theory. So all such theories are not modal at all. Finally, we obtain an extension theory for general quantum theories in characteristic 0 which allows one to extend any such theory over algebraically closed fields (such as classical complex quantum theory) to larger theories in which a quantum kernel is present. In this sense, these singular objects are always virtually around in abundance.
The usual description of the superposition of two (pure quantum) states is ambiguous, since the binary operation of addition in a Hilbert space does not pass down to the quotient projective space. In the setting of complex projective space this paper gives a new geometrical definition of the superposition of two pure states, viewed as two distinct points in the projective space, as the unique (complex) line on which those two points lie. Using the Erlangen program the geometry appropriate for quantum theory of the projective space is precisely defined. Also this rigorous approach is compared with others.