Finite Lorentz transformations, automorphisms, and division algebras

Manogue, Corinne A.; Schray, Jörg

doi:10.1063/1.530056

Cited by 77 publications

(170 citation statements)

References 14 publications

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“…In this case a Hilbert space formulation is not possible due to the nonassociative nature of the octonions. It has, however, been shown that the usual axioms of quantum theory may be satisfied by taking a more abstract "propositional" approach [194,214,217,219,220]. Let us compare J 3 (P ) of section 9.4 with the 2 qutrit reduced matrices ρ A and ρ B ρ A = Tr B |Ψ Ψ|, ρ B = Tr A |Ψ Ψ|, (13.30) which are also 3 × 3 hermitian and transform in the same way, at least in the R and C cases:…”

Section: The Jordan Algebra Formulation Of Quantum Mechanicsmentioning

confidence: 99%

Black holes, qubits and octonions

et al. 2009

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We review the recently established relationships between black hole entropy in string theory and the quantum entanglement of qubits and qutrits in quantum information theory. The first example is provided by the measure of the tripartite entanglement of three qubits (Alice, Bob and Charlie), known as the 3-tangle, and the entropy of the 8-charge ST U black hole of N = 2 supergravity, both of which are given by the [SL(2)] 3 invariant hyperdeterminant, a quantity first introduced by Cayley in 1845. Moreover the classification of three-qubit entanglements is related to the classification of N = 2 supersymmetric ST U black holes. There are further relationships between the attractor mechanism and local distillation protocols and between supersymmetry and the suppression of bit flip errors. At the microscopic level, the black holes are described by intersecting D3-branes whose wrapping around the six compact dimensions T 6 provides the string-theoretic interpretation of the charges and we associate the three-qubit basis vectors, |ABC (A, B, C = 0 or 1), with the corresponding 8 wrapping cycles. The black hole/qubit correspondence extends to the 56 charge N = 8 black holes and the tripartite entanglement of seven qubits where the measure is provided by Cartan's E 7 ⊃ [SL(2)] 7 invariant. The qubits are naturally described by the seven vertices ABCDEF G of the Fano plane, which provides the multiplication table of the seven imaginary octonions, reflecting the fact that E 7 has a natural structure of an O-graded algebra. This in turn provides a novel imaginary octonionic interpretation of the 56 = 7 × 8 charges of N = 8: the 24 = 3 × 8 NS-NS charges correspond to the three imaginary quaternions and the 32 = 4 × 8 R-R to the four complementary imaginary octonions. We contrast this approach with that based on Jordan algebras and the Freudenthal triple system. N = 8 black holes (or black strings) in five dimensions are also related to the bipartite entanglement of three qutrits (3-state systems), where the analogous measure is Cartan's E 6 ⊃ [SL(3)] 3 invariant. Similar analogies exist for magic N = 2 supergravity black holes in both four and five dimensions. Despite the ubiquity of octonions, our analogy between black holes and quantum information theory is based on conventional quantum mechanics but for completeness we also provide a more exotic one based on octonionic quantum mechanics. Finally, we note some intriguing, but still mysterious, assignments of entanglements to cosets, such as the 4-way entanglement of eight qubits to E 8 /[SL (2)] 8 .

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Section: The Jordan Algebra Formulation Of Quantum Mechanicsmentioning

confidence: 99%

Black holes, qubits and octonions

et al. 2009

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“…The following two lemmas are essentially contained in [38], but we would like to present a proof for the sake of completeness.…”

Section: Remarkmentioning

confidence: 99%

Plurisubharmonic Functions on the Octonionic Plane and Spin(9)-Invariant Valuations on Convex Sets

Alesker

2008

J Geom Anal

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“…More details specifically about the effects of the non-associativity of the octonions are given in [13,14].…”

Section: B Lorentz Transformationsmentioning

confidence: 99%

“…Only general properties of octonions independent of a specific multiplication table will be used here. However, because we make frequent use of a variety of octonionic identities, the reader may find more information on octonions helpful; see [14,15].…”

Section: A Octonionic Spinorsmentioning

confidence: 99%

The general classical solution of the superparticle

Schray¹

1996

Class. Quantum Grav.

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The theory of vectors and spinors in 9+1 dimensional spacetime is introduced in a completely octonionic formalism based on an octonionic representation of the Clifford algebra Cl(9, 1). The general solution of the classical equations of motion of the CBS superparticle is given to all orders of the Grassmann hierarchy. A spinor and a vector are combined into a 3 × 3 Grassmann, octonionic, Jordan matrix in order to construct a superspace variable to describe the superparticle. The combined Lorentz and supersymmetry transformations of the fermionic and bosonic variables are expressed in terms of Jordan products.

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Finite Lorentz transformations, automorphisms, and division algebras

Cited by 77 publications

References 14 publications

Black holes, qubits and octonions

Black holes, qubits and octonions

Plurisubharmonic Functions on the Octonionic Plane and Spin(9)-Invariant Valuations on Convex Sets

The general classical solution of the superparticle

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