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The Octonion Algebra in Quantum Mechanics

Hideki Sabata

Published Feb 28, 1997• 0 citations
OpenAlex

Abstract

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.

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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.
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.|この論文では, 量子力学における8元数代数の役割を研究することが目的とされている。最初の部分は8元数代数の公式を扱い, それから8元数のことばで量子力学を研究することが試みられる。 しかし, そのような試みが失敗することが示されている。しかし, この失敗は皮相的なことで, 2つの行列の積の定義によって失敗がもたらされるのではないかと思われる。 これを救う行列の積の定義の修正版が示唆される。
The present article is the flrst in a program that aims at generalizing quantum mechanics by keeping its structure essentially intact, but constructing the Hilbert space over a new number system much richer than the fleld of complex numbers. We call this number system \the Quantionic Algebra. It is eight dimensional like the algebra of octonions, but, unlike the latter, it is associative. It is not a division algebra, but \almost one (in a sense that will be evident when we come to it). It enjoys the minimum of properties needed to construct a Hilbert space that admits quantum-mechanical interpretations (like transition probabilities), and, moreover, it contains the local Minkowski structure of space-time. Hence, a quantum theory built over the quantions is inherently relativistic. The algebra of quantions has been discovered in two steps. The flrst is a careful analysis of the abstract structure of quantum mechanics (the flrst part of the present work), the second is the classiflcation of all concrete realizations of this abstract structure (several additional articles). The classiflcation shows that there are only two realizations. One is standard quantum mechanics, the other its inherently relativistic generalization. The present article develops the abstract algebra of observables.
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/
One of us has shown that the statistical properties of the measurements of a quantum mechanical system assume their simplest form when expressed in terms of a certain hypercomplex algebra which is commutative but not associative.1 This algebra differs from the non-commutative but associative matrix algebra usually considered in that one is concerned with the commutative expression ½(A × B + B × A) instead of the associative product A × B of two matrices. It was conjectured that the laws of this commutative algebra would form a suitable starting point for a generalization of the present quantum mechanical theory. The need of such a generalization arises from the (probably) fundamental difficulties resulting when one attempts to apply quantum mechanics to questions in relativistic and nuclear phenomena.
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.