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:2311.03647 · Journal of Physics A Mathematical and Theoretical

An algebraic formulation of nonassociative quantum mechanics

Peter Schupp, Richard J. Szabo

Published May 9, 2024• 3 citations
arXivHTMLPDF

Abstract

Abstract We develop a version of quantum mechanics that can handle nonassociative algebras of observables and which reduces to standard quantum theory in the traditional associative setting. Our algebraic approach is naturally probabilistic and is based on using the universal enveloping algebra of a general nonassociative algebra to introduce a generalized notion of associative composition product. We formulate properties of states together with notions of trace, and use them to develop Gel’fand–Naimark–Segal constructions. We describe Heisenberg and Schrödinger pictures of completely positive dynamics, and we illustrate our formalism on the explicit examples of finite-dimensional matrix Jordan algebras as well as the octonion algebra.

DOI
10.1088/1751-8121/ad4935

Related papers

Non-Associative Algebras and Quantum Physics -- A Historical Perspective

Manfred Liebmann, Horst Rühaak, Bernd Henschenmacher

We review attempts by Pascual Jordan and other researchers, most notably Lawrence Biedenharn to generalize quantum mechanics by passing from associative matrix or operator algebras to non-associative algebras. We start with Jordan's work from the early 1930ies leading to Jordan algebras and the first attempt to incorporate the alternative ring of octonions into physics. Jordan's work on the octonions from 1932 till 1952 will be covered, discussing aspects of the exceptional Jordan algebra and how to express probabilities when working with the octonions and the exceptional Jordan algebra. From the 1950ies onwards Jordan and others also considered one-sided distributive systems like near-fields, near-rings, quasi-fields and exceptional Segal systems (the last two examples being not necessarily associative). As the set of non-linear operators forms a near-ring and even a near-algebra, this may be of interest for attempts to pass from a linear to a non-linear setting in the study of quantum mechanics. Moreover, ideas introduced in the late 1960ies to use non-power-associative algebras to formulate a theory of a minimal length will be covered. Lawrence Biedenharn's and Jordan's ideas related to non-power-associative octonionic matrix algebras will be briefly mentioned, a long section is devoted to a summary of Horst Rühaak's PhD thesis from 1968 on matrix algebras over the octonions. Finally, recent attempts to use non-associative algebras in physics will be described.

arXiv:1909.04027•Sep 7, 2019•4 citations

Algebraic approach to entanglement and entropy

A. P. Balachandran, T. R. Govindarajan, Amilcar R. de Queiroz +1 more

We present a general approach to quantum entanglement and entropy that is based on algebras of observables and states thereon. In contrast to more standard treatments, Hilbert space is an emergent concept, appearing as a representation space of the observable algebra, once a state is chosen. In this approach, which is based on the Gel'fand-Naimark-Segal construction, the study of subsystems becomes particularly clear. We explicitly show how the problems associated with the partial trace for the study of entanglement of identical particles are readily overcome. In particular, a suitable entanglement measure is proposed that can be applied to systems of particles obeying Fermi, Bose, para-, and even braid-group statistics. The generality of the method is also illustrated by study of the time evolution of subsystems emerging from restriction to subalgebras. Problems related to anomalies and quantum epistemology are also discussed.

arXiv:1301.1300•Physical Review A•Aug 1, 2013•79 citations

The Mathematics of Quantum Mechanics

Jeffrey Alan Barrett

Abstract Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.

•Dec 26, 2019•1 citation

Testing Nonassociative Quantum Mechanics

Martin Bojowald A, Suddhasattwa Brahma, Umut Büyükçam

The familiar concepts of state vectors and operators in quantum mechanics rely on associative products of observables. However, these notions do not apply to some exotic systems such as magnetic monopoles, which have long been known to lead to nonassociative algebras. Their quantum physics has remained obscure. This Letter presents the first derivation of potentially testable physical results in nonassociative quantum mechanics, based on effective potentials. They imply new effects which cannot be mimicked in usual quantum mechanics with standard magnetic fields.

arXiv:1510.07559•Physical Review Letters•Nov 24, 2015•16 citations

Quantum Probability

Heinz‐Peter Breuer, Francesco Petruccione

Abstract This chapter develops the fundamental notions and concepts of the probabilistic and statistical interpretation of quantum mechanics. It begins with a brief review of the mathematical structure given by a Hilbert space of state vectors and an algebra of observables represented by selfadjoint operators. Special emphasis is laid on the connection between the spectral representation of a selfadjoint operator and a corresponding random variable that describes the possible measurement outcomes. The chapter further introduces the concepts of a composite quantum system, of the reduced density matrix, and of entangled quantum states. A section on quantum entropies is included, as well as the mathematical formulation of the generalized theory of quantum measurements in terms of completely positive quantum operations and effects.

•Jan 25, 2007•0 citations

Nonassociative quantum theory, emergent probability, and coquasigroup symmetry

Jens Köplinger, Джунушалиев Владимир Джумакадырович

This paper follows recent steps towards a nonassociative quantum theory and points out the mathematical structure behind the proposed modifications to conventional quantum theory. An N=1 supersymmetry model and a strong force glueball ansatz is highlighted. Using nonassociative complex octonion algebra, it is shown how the Lorentz Lie algebra can be understood as a four dimensional generalization of the algebra of spin-1/2 operators in physics. Probability is speculated to become an emergent phenomenon from some nonassociative geometry in which to better understand the fluxes involved. A prototype nonassociative quantum theory in one dimension is brought forward to illustrate how normed division algebras may aid in modeling isospin properties that are similar to observed field and particle symmetries in nature. This prototype is built from a principle of self-duality between types of active and passive transformations and supplied with a modified Born rule that models observation, similar to conventional quantum mechanics. Solutions on the complex numbers, quaternions and octonions are discussed. The Hopf coquasigroup structure of the octonionic eigenvalue relation is shown and advertised as a tool for future investigation into the complete solution set of the model.

arXiv:0910.3347•Oct 18, 2009•2 citations
Abstract We develop a version of quantum mechanics that can handle nonassociative algebras of observables and which reduces to standard quantum theory in the traditional associative setting. Our algebraic approach is naturally probabilistic and is based on using the universal enveloping algebra of a general nonassociative algebra to introduce a generalized notion of associative composition product. We formulate properties of states together with notions of trace, and use them to develop Gel’fand–Naimark–Segal constructions. We describe Heisenberg and Schrödinger pictures of completely positive dynamics, and we illustrate our formalism on the explicit examples of finite-dimensional matrix Jordan algebras as well as the octonion algebra.
We review attempts by Pascual Jordan and other researchers, most notably Lawrence Biedenharn to generalize quantum mechanics by passing from associative matrix or operator algebras to non-associative algebras. We start with Jordan's work from the early 1930ies leading to Jordan algebras and the first attempt to incorporate the alternative ring of octonions into physics. Jordan's work on the octonions from 1932 till 1952 will be covered, discussing aspects of the exceptional Jordan algebra and how to express probabilities when working with the octonions and the exceptional Jordan algebra. From the 1950ies onwards Jordan and others also considered one-sided distributive systems like near-fields, near-rings, quasi-fields and exceptional Segal systems (the last two examples being not necessarily associative). As the set of non-linear operators forms a near-ring and even a near-algebra, this may be of interest for attempts to pass from a linear to a non-linear setting in the study of quantum mechanics. Moreover, ideas introduced in the late 1960ies to use non-power-associative algebras to formulate a theory of a minimal length will be covered. Lawrence Biedenharn's and Jordan's ideas related to non-power-associative octonionic matrix algebras will be briefly mentioned, a long section is devoted to a summary of Horst Rühaak's PhD thesis from 1968 on matrix algebras over the octonions. Finally, recent attempts to use non-associative algebras in physics will be described.
We present a general approach to quantum entanglement and entropy that is based on algebras of observables and states thereon. In contrast to more standard treatments, Hilbert space is an emergent concept, appearing as a representation space of the observable algebra, once a state is chosen. In this approach, which is based on the Gel'fand-Naimark-Segal construction, the study of subsystems becomes particularly clear. We explicitly show how the problems associated with the partial trace for the study of entanglement of identical particles are readily overcome. In particular, a suitable entanglement measure is proposed that can be applied to systems of particles obeying Fermi, Bose, para-, and even braid-group statistics. The generality of the method is also illustrated by study of the time evolution of subsystems emerging from restriction to subalgebras. Problems related to anomalies and quantum epistemology are also discussed.
Abstract Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.
The familiar concepts of state vectors and operators in quantum mechanics rely on associative products of observables. However, these notions do not apply to some exotic systems such as magnetic monopoles, which have long been known to lead to nonassociative algebras. Their quantum physics has remained obscure. This Letter presents the first derivation of potentially testable physical results in nonassociative quantum mechanics, based on effective potentials. They imply new effects which cannot be mimicked in usual quantum mechanics with standard magnetic fields.
Abstract This chapter develops the fundamental notions and concepts of the probabilistic and statistical interpretation of quantum mechanics. It begins with a brief review of the mathematical structure given by a Hilbert space of state vectors and an algebra of observables represented by selfadjoint operators. Special emphasis is laid on the connection between the spectral representation of a selfadjoint operator and a corresponding random variable that describes the possible measurement outcomes. The chapter further introduces the concepts of a composite quantum system, of the reduced density matrix, and of entangled quantum states. A section on quantum entropies is included, as well as the mathematical formulation of the generalized theory of quantum measurements in terms of completely positive quantum operations and effects.
This paper follows recent steps towards a nonassociative quantum theory and points out the mathematical structure behind the proposed modifications to conventional quantum theory. An N=1 supersymmetry model and a strong force glueball ansatz is highlighted. Using nonassociative complex octonion algebra, it is shown how the Lorentz Lie algebra can be understood as a four dimensional generalization of the algebra of spin-1/2 operators in physics. Probability is speculated to become an emergent phenomenon from some nonassociative geometry in which to better understand the fluxes involved. A prototype nonassociative quantum theory in one dimension is brought forward to illustrate how normed division algebras may aid in modeling isospin properties that are similar to observed field and particle symmetries in nature. This prototype is built from a principle of self-duality between types of active and passive transformations and supplied with a modified Born rule that models observation, similar to conventional quantum mechanics. Solutions on the complex numbers, quaternions and octonions are discussed. The Hopf coquasigroup structure of the octonionic eigenvalue relation is shown and advertised as a tool for future investigation into the complete solution set of the model.