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

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Differences

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Either capital Latin letters (instead of lower case Latin letters), or Greek lower case, are used for indices. For example

 

other conventions...

Similarities

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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

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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

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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

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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

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Notes

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  1. ^ Van der Waerden B.L. (1929). "Spinoranalyse". Nachr. Ges. Wiss. Göttingen Math.-Phys. 1929: 100–109.
  2. ^ 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

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  • M. Carmeli, S. Malin (2000). Theory of Spinors. World Scientific. ISBN 9789812564726.
  • R. M. Wald (1984). General Relativity. Chicago University Press. ISBN 9780226870335. (spinor/twistor chapter, spinors of type (a,b;c,d)?)