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。
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。
There is a notational discrepancy in how mathematicians and physicists treat Hilbert spaces. In physics, one often works with finite-dimensional Hilbert spaces treated as ℂ n , and hence wants to write the Hermitian inner product as (x, y) = x ∗ y. This definition dictates the use of a Hermitian inner product that is antilinear in the first argument, the convention followed, for example, in Reed & Simon [246]. This is also the choice adopted in Dirac’s bracket notation, whose usage in quantum mechanics is very widespread. Our notation is chosen to extend-as closely as possible-the traditional notation of standard finite-dimensional matrix algebra to arbitrary complex inner product spaces and associated linear operators. In matrix algebra, column vectors and the corresponding matrices with one column are identical objects, row vectors are the linear functionals, and the adjoint is the conjugate transpose. For example, ℍ = ℍ × = ℂ n is the space of column vectors of size n, the dual space ℍ ∗ is the space of row vectors of size n, and the operator product ϕ ∗ ψ of a row vector ϕ ∗ and a column vector ψ is the standard Hermitian inner product of the column vectors ϕ and ψ. We use Greek lower case letters to write vectors, thus emphasizing their intended use as quantum state vectors in quantum mechanics. On the other hand, mathematicians working on reproducing kernel Hilbert spaces use an inner product (x, y) antilinear in the second argument, related to the physicist’s inner product (x, y) by (x, y) =( ). This is the convention followed, for example, in Rudin [252]. Although the two ways of defining the inner product lead to fully equivalent theories, all details look a bit different, a fact that has to be taken into account when reading the literature on the subject. For example, in the description based on the physical tradition, it is preferable to work with the antidual space in place of the dual space used in the mathematical tradition. In functional analysis, linear operators in Hilbert spaces are usually considered each with their own domain. But many computations in quantum mechanics require the consideration of algebras of operators with a common domain. The latter is a Euclidean space, a dense subspace of a Hilbert space. This space and its antidual play in many respects a more basic role in quantum physics than the Hilbert space itself. Therefore, and to avoid possible confusion caused by the different traditions, we give in the present chapter a self-contained introduction to Euclidean spaces and their associated spaces. All proofs are carried out in detail.
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.
The following notes introduce Quantum Mechanics at an advanced level addressing students of Physics, Mathematics, Chemistry and Electrical Engineering. The aim is to put mathematical concepts and techniques like the path integral, algebraic techniques, Lie algebras and representation theory at the readers disposal. For this purpose we attempt to motivate the various physical and mathematical concepts as well as provide detailed derivations and complete sample calculations. We have made every effort to include in the derivations all assumptions and all mathematical steps implied, avoiding omission of supposedly ‘trivial’ information. Much of the author’s writing effort went into a web of cross references accompanying the mathematical derivations such that the intelligent and diligent reader should be able to follow the text with relative ease, in particular, also when mathematically difficult material is presented. In fact, the author’s driving force has been his desire to pave the reader’s way into territories unchartered previously in most introductory textbooks, since few practitioners feel obliged to ease access to their field. Also the author embraced enthusiastically the potential of the TEX typesetting language to enhance the presentation of equations as to make the logical pattern behind the mathematics as transparent as possible. Any suggestion to improve the text in the respects mentioned are most welcome. It is obvious, that even though these notes attempt to serve the reader as much as was possible for the author, the main effort to follow the text and to master
FINDING: Non-associative quantum mechanics based on octonions and the exceptional Jordan algebra h₃(O) replaces linear Hilbert spaces with the octonionic projective plane, requiring a fundamentally non-associative algebraic foundation for quantum theory. MATH: - Octonions (O): 8-dimensional normed division algebra, non-associative, alternative (x(xy) = (xx)y, (yx)x = y(xx)). - Exceptional Jordan algebra h₃(O): 27-dimensional algebra of 3×3 Hermitian matrices over O, with product A∘B = ½(AB+BA). - Octonionic projective plane OP²: points correspond to rank-1 idempotents in h₃(O); dimension 16, automorphism group F₄ (exceptional Lie group). - Key constants: No direct ratios (0.382, 0.618, etc.) appear; instead, the structure is governed by the octonion multiplication table (Fano plane) and the 7-sphere S⁷. CONNECTION: - The Fano plane (projective plane of order 2) underlies octonion multiplication — a finite geometry with 7 points and 7 lines, each line containing 3 points. Th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
I start by briefly overviewing quantum mechanics as it is standardly formulated and by discussing the question of whether this standard formulation needs to be supplemented by an interpretation. The overview is based mainly on Von Neumann (1932, 1955) and its aim is not to give the reader a crash course in quantum mechanics, but to present those parts of the standard formalism which I use in this book. For a more complete treatment, the reader may consult the standard textbooks on quantum mechanics or, for instance, Sudbery (1986) or Redhead (1987). The standard formulation The standard formulation of quantum mechanics can be introduced in four steps. The first step is that in quantum mechanics one describesthe physics of a system by means of a Hilbert space ℋ. This Hilbert space is a complex linear vector space on which an inner product is defined. Let's adopt the convention (please consult the Glossary at the end of the book for notational conventions) that α refers to a system and that is the Hilbert space that is associated with this system. Let ∣ℋ α 〉 denote a vector in ℋ α and let 〈ψ α ∣φ α 〉 be the inner product between the vectors ∣ψ α 〉 and ∣φ α 〉.