In order to obtain a consistent formulation of octonionic quantum mechanics (OQM), we intro· duce left-right barred operators. Such operators enable us to find the translation rules between octonionic numbers and 8 x 8 real matrices (a translation is also given for 4 x 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. § 1. Introduction 833 Since the sixties, there has been renewed and intense interest in the use of octonions in physics. I) Octonionic algebra has been in fact linked with a number of interesting subjects: structure of interactions,2) SU(3) color symmetry and quark confinement,3),4) standard model gauge group,5) exceptional GUT groups,6) DiracClifford algebra/) nonassociative Yang-Mills theories,8),9) space-time symmetries in ten dimensions,IO) and supersymmetry and supergravity theories.ll ),12) Moreover, the recent successful application of quaternionic numbers in quantum mechanics/ 3H7 ) in particular in formulating a quaternionic Dirac equation/ 8 )-21) suggests going one step further and using octonions as an underlying numerical field.In this work, we overcome the problems due to the nonassociativity of the octonionic algebra by introducing left-right barred operators (which will be sometimes called barred octonions). Such operators complete the mathematical material introduced in the recent papers of Joshi et al. 8 ),9) Then, we investigate their relations to GL(8, !R.) and GL (4, C). Establishing this relation we find interesting translation rules, which give us the opportunity to formulate a consistent OQM.The philosophy behind the translation can be concisely expressed by the following statement: "There exists at least one version of octonionic quantum mechanics where the standard quantum mechanics is reproduced". The use of a complex scalar product (complex geometry)22) will be the main tool to obtain OQM.We wish to stress that translation rules do not imply that our octonionic quantum world (with complex geometry) is equivalent to the standard quantum world. When translation fails the two worlds are not equivalent. An interesting case can be supersymmetry.23) Similar translation rules, between quaternionic quantum mechanics (QQM) with complex geometry and standard quantum mechanics, have recently been found. 16 ) As an application, such rules can be exploited in reformulating in a natural way of the electroweak sector of the standard model. 17 )In § 2, we discuss octonionic algebra and introduce barred operators. Then, in
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 details.
Octonionic algebra being nonassociative is difficult to manipulate. We introduce left-right octonionic barred operators which enable us to reproduce the associative GL(8, R) group. Extracting the basis of GL(4, C), we establish an interesting connection between the structure of left-right octonionic barred operators and generic 4 × 4 complex matrices. As an application we give an octonionic representation of the 4-dimensional Clifford algebra.PACS numbers: 02.10.Jf , 02.10.Vr , 02.20.Qs . KeyWords: octonions, barred operators, Clifford algebra. I. INTRODUCTIONSemi-simple Lie groups, classified in four categories: orthogonal groups, unitary groups, symplectic groups and exceptional groups, were respectively associated with real, complex, quaternionic and octonionic algebras. Thus, such algebras became the core of the classification of possible symmetries in physics [1][2][3][4].We know that the antihermitian generators of SU (2, C) can be represented by the three quaternionic imaginary units e 1 , e 2 , e 3It permits any quaternionic numbers or matrix to be translated into a complex matrix but not necessarily vice-versa. In fact, to define the most general 2×2 complex matrix, we need 8 real numbers. This problem is solved by introducing the barred quaternion 1 | e 1 (↔ i1 1 2×2 ) which allows to obtain a faithful quaternionic representation of GL(2, C) [5]. Exploiting the barred operator idea, we find the following 16 quaternionic operatorswhere Q ≡ (e 1 , e 2 , e 3 ). These operators become essential to formulate special relativity with real quaternions [6], allowing to overcome the difficulties which in the past did not permit a (real) quaternionic version of special relativity. Besides, they can be used to give a representation of GL(4, R). The situation can be summarized as followswith q, p, r, s quaternionic numbers.Inspiring by this sequence we try to extend it and find an isomorphism between octonions and 8 × 8 real [or 4 × 4 complex] matrices. Obviously a first difficulty is the following: The octonionic algebra is nonassociative whereas GL(8, R) [or GL(4, C)], satisfying the Jacobi identity, is associative. This seems a hopeless situation.In this paper, we introduce left/right octonionic barred operators which enables us to find translation rules between 8×8 real matrices and octonionic numbers. On our road we also find an interesting isomorphism between the structure of left/right octonionic barred operators, on the one hand, and 4 × 4 complex matrices, on the other hand.This article is organized as follows: In section II, we give a brief introduction to the octonionic division algebra. In section III, we discuss octonionic barred operators and explain the need to distinguish between left-barred and rightbarred operators. In section IV, we investigate the relation between barred octonions and 8 × 8 real matrices. In this section, we also give the translation rules between our octonionic barred operators and GL(4, C) and as an application *
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