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

Pushpa, K.; Bisht, P. S.; Negi, O. P. S.

doi:10.1007/s10773-014-2022-z

Cited by 5 publications

(4 citation statements)

References 15 publications

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“…This ubiquitous appearance of three entities at a vertex in physics has been suggestive of connections to other areas of mathematics. Thus, quaternionic and octonionic symmetries for braids on three strands, their group structure, and triplets of half-odd integer labels have been discussed for the standard model of particle physics [105] and quantum gravity [106], as also the closely related study of knots [107]; see also [108]. A book on division algebras in particle physics is additionally helpful [109].…”

Section: Geometric Viewmentioning

confidence: 99%

Symmetries and Geometries of Qubits, and Their Uses

Rau

2021

Symmetry

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The symmetry SU(2) and its geometric Bloch Sphere rendering have been successfully applied to the study of a single qubit (spin-1/2); however, the extension of such symmetries and geometries to multiple qubits—even just two—has been investigated far less, despite the centrality of such systems for quantum information processes. In the last two decades, two different approaches, with independent starting points and motivations, have been combined for this purpose. One approach has been to develop the unitary time evolution of two or more qubits in order to study quantum correlations; by exploiting the relevant Lie algebras and, especially, sub-algebras of the Hamiltonians involved, researchers have arrived at connections to finite projective geometries and combinatorial designs. Independently, geometers, by studying projective ring lines and associated finite geometries, have come to parallel conclusions. This review brings together the Lie-algebraic/group-representation perspective of quantum physics and the geometric–algebraic one, as well as their connections to complex quaternions. Altogether, this may be seen as further development of Felix Klein’s Erlangen Program for symmetries and geometries. In particular, the fifteen generators of the continuous SU(4) Lie group for two qubits can be placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups. The very different perspectives that we consider may provide further insight into quantum information problems. Extensions are considered for multiple qubits, as well as higher-spin or higher-dimensional qudits.

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Section: Geometric Viewmentioning

confidence: 99%

Symmetries and Geometries of Qubits, and Their Uses

Rau

2021

Symmetry

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“…In physical applications, split octonions were used to provide possible explanation for the existence of three generations of fermionic elementary particles [41] [42]. In [43] generators of ( )…”

Section: Introductionmentioning

confidence: 99%

Dirac and Maxwell Systems in Split Octonions

Gogberashvili¹,

Gurchumelia²

2023

JAMP

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The known equivalence of 8-dimensional chiral spinors and vectors, also referred to as triality, is discussed for (4 + 4)-space. Split octonionic representation of SO(4, 4) and Spin(4, 4) groups and the trilinear invariant form are explicitly written and compared with Clifford algebraic matrix representation. It is noted that the complete algebra of split octonionic basis units can be recovered from the Moufang and Malcev relations for the three vector-like elements. Lagrangians on split octonionic fields that generalize Dirac and Maxwell systems are constructed using group invariant forms. It is shown that corresponding equations are related to split octonionic analyticity conditions.

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“…Split octonions were used to provide possible explanation for the existence of three generations of fermionic elementary particles [40,41]. In [42] generators of SO (8) and SO(7) groups were obtained and have been used to describe the rotational transformation in 7-dimensional space. In [43][44][45] real reducible 16 × 16-matrix representation of SO(4, 4) group utilizing the Clifford algebra approach was constructed and it was shown that there are two inequivalent real 8 × 8 irreducible basic spinor representations, potential physical applications for 8-dimensional electrodynamics [44] and gravity [45] was also considered.…”

Section: Introductionmentioning

confidence: 99%

Split Octonions and Triality in (4+4)-Space

Gogberashvili¹,

Gurchumelia²

2020

Preprint

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The known equivalence of 8-dimensional chiral spinors and vectors is discussed for (4+4)space within the context of the algebra of the split octonions. It is shown that the complete algebra of hyper-complex octonionic basis units can be recovered from the Moufang and Malcev relations for the three vector-like elements of the split octonions. Trilinear form, which is invariant under SO(4,4) transformations for vectors and corresponding Spin(4,4) transformations for spinors, is explicitly written using both purely matrix and purely octonionic representations.

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

Cited by 5 publications

References 15 publications

Symmetries and Geometries of Qubits, and Their Uses

Symmetries and Geometries of Qubits, and Their Uses

Dirac and Maxwell Systems in Split Octonions

Split Octonions and Triality in (4+4)-Space

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