User:Maschen/spinor index notation
In mathematics and mathematical physics, spinor index notation is the index notation analogue of tensor index notation for spinors. Many of the tensor index notation conventions carry over to spinor index notation.
Overlap with tensor index notation
[edit]Differences
[edit]Either capital Latin letters (instead of lower case Latin letters), or Greek lower case, are used for indices. For example
other conventions...
Similarities
[edit]- Summation convention for twice-repeated indices
- Covariance
Lower indices, ψμν...
- Contravariance
Upper indices, ψαβ...
- Mixed variance
Upper and lower indices ψαμ.
Raising and lowering indices via the metric spinor also applies, e.g.
- Symmetry
- Antisymmetry
van der Waerden notation
[edit]In theoretical physics, van der Waerden notation[1][2] refers to the usage of two-component spinors (Weyl spinors) in four spacetime dimensions. This is standard in twistor theory and supersymmetry. It is named after Bartel Leendert van der Waerden.
Dots on indices
[edit]- Undotted indices (chiral indices)
Spinors with lower undotted indices have a left-handed chiralty, and are called chiral indices.
- Dotted indices (anti-chiral indices)
Spinors with raised dotted indices, plus an overbar on the symbol (not index), are right-handed, and called anti-chiral indices.
Without the indices, i.e. "index free notation", an overbar is retained on right-handed spinor, since ambiguity arises between chiralty when no index is indicated.
Hats on indices
[edit]Indices which have hats are called Dirac indices, and are the set of dotted and undotted, or chiral and anti-chiral, indices. For example, if
then a spinor in the chiral basis is represented as
where
In this notation the Dirac adjoint (also called the Dirac conjugate) is
See also
[edit]- Dirac equation
- bra–ket notation
- Infeld–van der Waerden symbols
- Lorentz transformation
- Pauli equation
- Ricci calculus
Notes
[edit]- ^ Van der Waerden B.L. (1929). "Spinoranalyse". Nachr. Ges. Wiss. Göttingen Math.-Phys. 1929: 100–109.
- ^ Veblen O. (1933). "Geometry of two-component Spinors". Proc. Natl. Acad. Sci. USA. 19 (4): 462–474. Bibcode:1933PNAS...19..462V. doi:10.1073/pnas.19.4.462. PMC 1086023. PMID 16577541.
References
[edit]- M. Carmeli, S. Malin (2000). Theory of Spinors. World Scientific. ISBN 9789812564726.
- Landau, L.D.; Lifshitz, E. M. (1977). Quantum Mechanics: Non-Relativistic Theory. Vol. Vol. 3 (3rd ed.). Pergamon Press. ISBN 978-0-08-020940-1.
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has extra text (help) Online copy
- R. M. Wald (1984). General Relativity. Chicago University Press. ISBN 9780226870335. (spinor/twistor chapter, spinors of type (a,b;c,d)?)
- Spinors in physics
- P. Labelle (2010), Supersymmetry, Demystified series, McGraw-Hill (USA), ISBN 978-0-07-163641-4
- Hurley, D.J.; Vandyck, M.A. (2000), Geometry, Spinors and Applications, Springer, ISBN 1-85233-223-9
- Penrose, R.; Rindler, W. (1984), Spinors and Space–Time, vol. Vol. 1, Cambridge University Press, ISBN 0-521-24527-3
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has extra text (help) - Budinich, P.; Trautman, A. (1988), The Spinorial Chessboard, Springer-Verlag, ISBN 0-387-19078-3