A Non-Associative Quantum Mechanics

Abstract

A non-associative quantum mechanics is proposed in which the product of three and more operators can be non-associative one. The multiplication rules of the octonions define the multiplication rules of the corresponding operators with quantum corrections. The self-consistency of the operator algebra is proved for the product of three operators. Some properties of the non-associative quantum mechanics are considered. It is proposed that some generalization of the non-associative algebra of quantum operators can be helpful for understanding of the algebra of field operators with a strong interaction.

Role of division algebra in seven-dimensional gauge theory

“…Rotation R 12 yields three simultaneous rotations by an angle 2θ in three planes (1,2), (5,4) and (6,7) and leaves x 3 unchanged. The above transformations of x under R 12 can be written as…”

“…(19), J 3 is 7 × 7 matrix which represents rotation around the x 3 axis or in the (1, 2), (5,4) and (6,7) planes, and it can be written as…”

“…The algebra of octonions O defined on the eight-dimensional real vector space R 8 and is a non-associative extension of the algebra of quaternions H. Octonion represents an alternative (the alternation of associator under the permutation of arguments) and composition (it satisfy the property N (xy) = N (x)N (y) for the norm N (y) = yȳ =ȳy; x, y ∈ O) algebra. These composition algebras 2 are responsible for many important mathematical structures of interest for physicists in various ways to describe properties of electromagnetism, 5 quantum mechanics, 6 field theory, 7 gauge theory, [8][9][10] P. Kalauni & J. C. A. Barata gravitation, 11,12 and they have various applications in seven-dimensional space [13][14][15] as well. Various applications of octonion have been widely discussed by Baez.…”

Reconstruction of symmetric Dirac–Maxwell equations using nonassociative algebra

“…Octonions form eight-dimensional algebra over real numbers, it is noncommutative as well as nonassociative. Octonions have been used in various ways to describe properties of quantum mechanics [13][14][15], field theory [16,17], gauge theory [18][19][20] and Dirac equations [21,22]. Recently a great interest has been paid in the connection of octonions with electromagnetism [23][24][25][26][27].…”

Generalised Split Octonions and Their Transformation in SO(7) Symmetry

“…Lassig and Joshi [7] introduced the bi-modular representation of octonions and formulated the SO(8) gauge theory equivalent to the octonionic construction, also the peculiarities of the eight dimensional space has been described by Gamba [8]. Octonionic description has been used to describe the various applications in quantum chromo dynamics [9], in the study of representations of Clifford algebras [10], non associative quantum mechanics [11], in the study of symmetry breaking [12], in the study of flavor symmetry [13], in proposals of unified field theory models [14], unitary symmetry [15], octonionic gravity [16]. In the present study, we have described the generalized split octonionic description to represent the 8-dimensional algebra.…”