arXiv:hep-th/9609032 · Progress of Theoretical Physics

Octonionic Quantum Mechanics and Complex Geometry

Stefano De Leo, Khaled Abdel-Khalek

Published Oct 1, 1996• 36 citations

Abstract

The use of complex geometry allows us to obtain a consistent formulation of octonionic quantum mechanics (OQM). In our octonionic formulation we solve the hermiticity problem and define an appropriate momentum operator within OQM. The nonextendability of the completeness relation and the norm conservation is also discussed in detail.

DOI: 10.1143/ptp.96.823

Related papers

The octonionic eigenvalue problem

Stefano De Leo, Gisele Cristina Ducati

By using a real matrix translation, we propose a coupled eigenvalue problem for octonionic operators. In view of possible applications in quantum mechanics, we also discuss the hermiticity of such operators. Previous difficulties in formulating a consistent octonionic Hilbert space are solved by using the new coupled eigenvalue problem and introducing an appropriate scalar product for the probability amplitudes.

arXiv:1208.4999Journal of Physics A Mathematical and TheoreticalJul 23, 201215 citations

Octonionic Dirac Equation

Stefano De Leo, Khaled Abdel-Khalek

In order to obtain a consistent formulation of octonionic quantum mechanics (OQM), we introduce left-right barred operators. Such operators enable us to find the translation rules between octonionic numbers and 8×8 real matrices (a translation is also given for 4×4 complex matrices). We develop an octonionic relativistic free wave equation, linear in the derivatives. Even if the wave functions are only one-component we show that four independent solutions, corresponding to those of the Dirac equation, exist.

arXiv:hep-th/9609033Progress of Theoretical PhysicsOct 1, 199637 citations

Quantum Mechanics from the Constraint Axiom: A Derivation from Octonionic Algebra

Murad Ahmadov

The complete mathematical structure of quantum mechanics is derived from first principles, tracing all fundamental results to the constraint axiom via octonionic algebra. Without postulates, empirical fitting, or measurement assumptions, the following results are established: (1) Quantum states must reside in a complex Hilbert space over the complex numbers. (2) The canonical commutation relation between position and momentum follows from octonionic non-commutativity together with holographic information bounds. (3) The Heisenberg uncertainty principle arises as a mathematical theorem from the Cauchy–Schwarz inequality. (4) The Schrödinger equation emerges from time translation symmetry. (5) Spin one-half arises from the structure of the relevant Clifford algebra. (6) The Pauli matrices originate from the quaternion subalgebra contained within the octonions. (7) The Born rule appears as the unique probability measure on complex projective space. Related resourcesAdditional preprints, theoretical frameworks, and ongoing work by the author are available at:https://murad-ahmadov.github.io/

Mar 19, 20260 citations

An ℰ6⊗𝒰(1) invariant quantum mechanics for a Jordan pair

Piero Truini, Lawrence C. Biedenharn

Quantum mechanical spaces associated with geometries based on exceptional groups are of interest as models for internal (quark) symmetries. Using the concept of a Jordan pair, two copies of complex 3×3 octonionic Jordan algebras (ℳ83) are shown to define a quantum mechanics over the complex octonionic plane having ℰ6⊗𝒰(1) as automorphism group. The unusual features of this new quantal structure (neither a projective geometry, nor a lattice) are discussed.

Journal of Mathematical PhysicsJul 1, 198224 citations

Geometric formulation of quantum mechanics

Hoshang Heydari

Quantum mechanics is among the most important and successful mathematical model for describing our physical reality. The traditional formulation of quantum mechanics is linear and algebraic. In contrast classical mechanics is a geometrical and non-linear theory that is defined on a symplectic manifold. However, after invention of general relativity, we are convinced that geometry is physical and effect us in all scale. Hence the geometric formulation of quantum mechanics sought to give a unified picture of physical systems based on its underling geometrical structures, e.g., now, the states are represented by points of a symplectic manifold with a compatible Riemannian metric, the observables are real-valued functions on the manifold, and the quantum evolution is governed by a symplectic flow that is generated by a Hamiltonian function. In this work we will give a compact introduction to main ideas of geometric formulation of quantum mechanics. We will provide the reader with the details of geometrical structures of both pure and mixed quantum states. We will also discuss and review some important applications of geometric quantum mechanics.

arXiv:1503.00238Mar 1, 201516 citations

The Octonion Algebra in Quantum Mechanics

Hideki Sabata

In this article, the author aims to study the role of the octonion algebra in quantum mechanics. First part treats the formulae of octonion algebra and then we try to study quantum mechanics in the terms of octonion algebra and show that we fall to fail in that trial. But this failure may be only superficial which comes from the usual definition of the product of two matrices. We suggest the correct definition of the product of matrices.

Feb 28, 19970 citations