Class of spinors constructed using Clifford algebras
In the domain of mathematics known as representation theory, pure spinors (or simple spinors) are spinors that are annihilated, under the Clifford algebra representation, by a maximal isotropic subspace of a vector space with respect to a scalar product .
They were introduced by Élie Cartan[1] in the 1930s and further developed by Claude Chevalley.[2]
They are a key ingredient in the study of spin structures and higher dimensional generalizations of twistor theory,[3] introduced by Roger Penrose in the 1960s.
They have been applied to the study of supersymmetric Yang-Mills theory in 10D,[4][5] superstrings,[6] generalized complex structures[7][8] and parametrizing solutions of integrable hierarchies.[9][10][11]
^Cartan, Élie (1981) [1938]. The theory of spinors. New York: Dover Publications. ISBN 978-0-486-64070-9. MR 0631850.
^Chevalley, Claude (1996) [1954]. The Algebraic Theory of Spinors and Clifford Algebras (reprint ed.). Columbia University Press (1954); Springer (1996). ISBN 978-3-540-57063-9.
^Penrose, Roger; Rindler, Wolfgang (1986). Spinors and Space-Time. Cambridge University Press. pp. Appendix. doi:10.1017/cbo9780511524486. ISBN 9780521252676.
^Witten, E. (1986). "Twistor-like transform in ten dimensions". Nuclear Physics. B266 (2): 245–264. Bibcode:1986NuPhB.266..245W. doi:10.1016/0550-3213(86)90090-8.
^Harnad, J.; Shnider, S. (1986). "Constraints and Field Equations for Ten Dimensional Super Yang-Mills Theory". Commun. Math. Phys. 106 (2): 183–199. Bibcode:1986CMaPh.106..183H. doi:10.1007/BF01454971. S2CID 122622189.
^Cite error: The named reference Berk was invoked but never defined (see the help page).
^Gualtieri, Marco (2011). "Generalized complex geometry". Annals of Mathematics. (2). 174 (1): 75–123. arXiv:0911.0993. doi:10.4007/annals.2011.174.1.3.
^Date, Etsuro; Jimbo, Michio; Kashiwara, Masaki; Miwa, Tetsuji (1982). "Transformation groups for soliton equations IV. A new hierarchy of soliton equations of KP type". Physica. 4D (11): 343–365.
^Date, Etsuro; Jimbo, Michio; Kashiwara, Masaki; Miwa, Tetsuji (1983). M. Jimbo and T. Miwa (ed.). "Transformation groups for soliton equations". In: Nonlinear Integrable Systems - Classical Theory and Quantum Theory. World Scientific (Singapore): 943–1001.
^Balogh, F.; Harnad, J.; Hurtubise, J. (2021). "Isotropic Grassmannians, Plücker and Cartan maps". Journal of Mathematical Physics. 62 (2): 121701. arXiv:2007.03586. doi:10.1063/5.0021269. S2CID 220381007.
the spinor representation on the irreducible spinor (or half-spinor) modules. Purespinors, defined up to projectivization, are called projective pure spinors...
tensors, a spinor transforms to its negative when the space rotates through 360° (see picture). It takes a rotation of 720° for a spinor to go back to...
algebra on spinors. A spinor is said to be a purespinor if it is annihilated by half of a set of a set of generators of the Clifford algebra. Spinors are sections...
projective purespinor field. If M is a spin manifold, then Hol(ω) ⊂ SU(n) if and only if M admits at least two linearly independent parallel purespinor fields...
S2CID 53666173. Berkovits, Nathan (2014-03-01). "Infinite tension limit of the purespinor superstring". Journal of High Energy Physics. 2014 (3): 17. arXiv:1311...
of spinning string, maintaining space-time supersymmetry manifest, is possible in a formalism inspired on the GS formalism, known as purespinor formalism...
deformations. A formulation of superstring field theory using the non-minimal pure-spinor variables was introduced by Berkovits. The action is cubic and includes...
are not eigenfunctions of the total spin-squared operator, Ŝ2, but can formally be expanded in terms of purespin states of higher multiplicities (the...
the abstract vector states. In both categories, quantum states divide into pure versus mixed states, or into coherent states and incoherent states. Categories...
this half-dimensional space is the annihilator of a nowhere vanishing purespinor then M is a generalized Calabi–Yau manifold. Almost quaternionic manifold –...
real dimension of the pure state space of an m-qubit quantum register is 2m+1 − 2. Mathematically the Bloch sphere for a two-spinor state can be mapped...
Spin pumping is the dynamical generation of purespin current by the coherent precession of magnetic moments, which can efficiently inject spin from a...
projectivized minimal spinor orbit, under the spin representation, the so-called projective purespinor variety which, similarly to the image of the Plücker...
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used in some parts of geometry. See: Spin group Spin-c group Spinor Pin group Pinors Spinor field Killing spinorSpin manifold Ricci, Gregorio; Levi-Civita...
This effect is similar to the spin Hall effect, where a purespin current is induced by an electrical current. The spin Nernst effect can be detected...
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on superstring field theory, twistor string theory, topological and pure-spinor strings. He obtained exact solutions to Berkovits’ string field theory...
which are (maximally) entangled pure states (pure states of the HA ⊗ HB space, but which cannot be separated into pure states of each HA and HB). Now suppose...
In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product...
matrices In pure mathematics and physics: Wigner D-matrix, represent spins and rotations of quantum states and tensor operators. Higher spin alternating...
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