theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...

, Van der Waerden notation refers to the usage of two-component

spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...

s (

Weyl spinor In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...

s) in four spacetime dimensions. This is standard in

twistor theory In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a branch of theoretical and mathematical physics. Penrose proposed that twistor space should be the basic are ...

and supersymmetry. It is named after

Bartel Leendert van der Waerden Bartel Leendert van der Waerden (; 2 February 1903 – 12 January 1996) was a Dutch mathematician and historian of mathematics. Biography Education and early career Van der Waerden learned advanced mathematics at the University of Amster ...

Dotted indices

;Undotted indices (chiral indices) Spinors with lower undotted indices have a left-handed chirality, and are called chiral indices. :

\Sigma_\mathrm = 
\begin
\psi_\\
0
\end

;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. :

\Sigma_\mathrm = 
\begin
0 \\
\bar^\\
\end

Without the indices, i.e. "index free notation", an overbar is retained on right-handed spinor, since ambiguity arises between chirality when no index is indicated.

Hatted indices

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 :

\alpha = 1,2\,,\dot = \dot,\dot

then a spinor in the chiral basis is represented as :

\Sigma_\hat = 
\begin
\psi_\\
\bar^\\
\end

where :

\hat= (\alpha,\dot) = 1,2,\dot,\dot

In this notation the

Dirac adjoint In quantum field theory, the Dirac adjoint defines the dual operation of a Dirac spinor. The Dirac adjoint is motivated by the need to form well-behaved, measurable quantities out of Dirac spinors, replacing the usual role of the Hermitian adjoi ...

(also called the Dirac conjugate) is :

\Sigma^\hat = 
\begin
\chi^ & \bar_
\end

Notes

References

Spinors in physics
* * * * {{tensors Spinors Mathematical notation

Dotted indices

Hatted indices

See also

Notes

References