Clifford Algebras and Spinors

Lounesto, Pertti

doi:10.1007/978-94-009-4728-3_2

Cited by 457 publications

(887 citation statements)

References 9 publications

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“…The group of operations thus defined is the P in(2N) group, the double cover of O(2N) [21,22,23]. In turn, the subgroup of P in(2N) generated by an even number of reflections is the covering group of SO(2N), Spin(2N), associated with the linear fermion canonical transformations (also known as the Bogoliubov-Valatin [25] or squeeze transformations [26]).…”

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confidence: 99%

BCS-like modewise entanglement of fermion Gaussian states

Botero

Reznik

2004

Physics Letters A

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We show that with respect to any bipartite division of modes, pure fermion gaussian states display the same type of structure in its entanglement of modes as that of the BCS wave function, namely, that of a tensor product of entangled two-mode squeezed fermion states. We show that this structure applies to a wider class of "isotropic" mixed fermion states, for which we derive necessary and sufficient conditions for mode entanglement.The nature of many-body entanglement in various solid-state models such as spin chains, superconductivity, and harmonic chains, has been a recent subject of intense investigation [1][2][3][4][5][6][7][8][9][10][11][12]. The motivation for this effort is two-fold: on the one hand, such systems exhibit a rich entanglement structure which could potentially be harnessed for quantum information processing [7,8,9,10]; on the other hand, this effort promises to deepen our understanding of certain universal features in many-body physics-such as quantum phase transitions-and their relation to entanglement [2][3][4][5]12].In this Letter we investigate the properties of bipartite fermion mode entanglement [1] for a wide class of models describing correlated fermion systems, and show that this entanglement displays a simple, universal structure. As it is well known, under certain field transformations or after suitable approximations, a large class of interacting theories can be * Electronic address: abotero@uniandes.edu.co † Electronic address: reznik@post.tau.ac.il

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confidence: 99%

BCS-like modewise entanglement of fermion Gaussian states

Botero

Reznik

2004

Physics Letters A

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“…It is important to choose a proper framework for model building and a number system which can incorporate the complexity of Nature. In the expression of fiber bundles (E, Π ,F, G, X) [9][10][11][12], Connes defined two fibers F that are extended from a point of a base space X = S 3 , and the group G that defines dynamics of leptons described by Dirac equation is expressed by quaternions ∈ H, and a tangential space TS 3 was identified with S 3 × R 4 spacetime.…”

Section: Hmentioning

confidence: 99%

“…Induced representations [10] are useful in describing a group G which hasa closed normal subgroup N. Irreducible representations of semi-direct product G is written in the form nh.…”

Section: Hmentioning

confidence: 99%

E. Cartan’s and A. Connnes’ Supersymmetry and Differences of Understanding Physical Phenomena

Furui¹

2018

JMSS

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Einstein claimed that one cannot define global time, and proposed defining local time additionally. Such approach was adopted also by E. Cartan, in which fermions are described by spinors with 16 bases and interact with vectors with 8 bases, that consists of a couple of 4 dimensional vectors x i (i = 1, ···, 4) and x i (i = 1, ···, 4). In Cartan's theory, spinors and vectors transform by super symmetric transformations G 23 , G 12 , G 13 , G 123 and G 132 and bases of fermion spinors consist of ξ 0 ,ξ i (i = 1, ···, 4), ξ 1234 , ξ 234 , ξ 134 , ξ 124 , ξ 123 and ξ i,j (i /= j ∈{1,2,3,4}). Except G 23 , the transformations mix spinors and vectors, and operations of G 23 on spinors contain G 23 ξ 4 = ξ 0 and G 23 ξ 123 = ξ 1234 , and operations of G 23 on vectors contain G 23 x 4 = −x 4' and G 23 x 4' = −x 4 . Therefore, there are 14 independent spinor bases and 7 independent vector bases, which corresponds to the number of bases of the G 2 symmetry.From the bases of non-commutative geometry, Connes took two fibers from a point of S 3 basis, and on top of fibers allowed two times propagate following von Neumann algebra, but evolution of the system was assumed to be defined by one-parameter group of transformation. Steenrod stated that the S 7 symmetry can be regarded as S 3 symmetry covered over S 4 symmetry, which allows decomposition of S

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“…One should note that even four-vectors cannot achieve this kind of union: the spatial vector component of four-vectors belongs to ordinary three-dimensional space and thus is not properly connected to a specific observer (with a specific proper time). On the other hand, the dyadic notation and the vector analysis in ordinary threedimensional space, introduced by Gibbs, suffer from a major flaw: the cross product of vectors, apart from not being associative, can only be defined in three or seven dimensions [11]. Spacetime algebra [12], the geometric algebra [13,14] of spacetime, is -from the authors' viewpoint -the most adequate formalism to achieve that union of space and time envisioned by Minkowski.…”

Section: Introductionmentioning

confidence: 99%

Doppler Shift from a Composition of Boosts with Thomas Rotation: a Spacetime Algebra Approach

Paiva¹,

Ribeiro²

2006

Journal of Electromagnetic Waves and Applications

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Abstract-As is well-known, Lorentz boosts do not form a group: the composite of two boosts is, in general, equivalent to the composite of a rotation -the so-called Thomas (or Wigner) rotation -and a single boost. Using spacetime algebra, the geometric algebra for Minkowski spacetime, we show in this paper how to relate the Thomas rotation to the Doppler shift experienced between two observers in relative motion. Both observers are moving away from a third observer, set in the laboratory, with relative non-parallel velocities.

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Clifford Algebras and Spinors

Cited by 457 publications

References 9 publications

BCS-like modewise entanglement of fermion Gaussian states

BCS-like modewise entanglement of fermion Gaussian states

E. Cartan’s and A. Connnes’ Supersymmetry and Differences of Understanding Physical Phenomena

Doppler Shift from a Composition of Boosts with Thomas Rotation: a Spacetime Algebra Approach

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