Bispinor

In physics, bispinor is a four-component object which transforms under the (½,0)⊕(0,½) representation of the covariance group of special relativity [covariance group of special relativity is $O^+(1,3)$ or the Lorentz group.] (see, e.g., [Caban and Rembielinski 2005, p. 2.] ). Bispinors are used to describe relativistic spin-½ quantum fields.

In the Weyl basis a bispinor:consists of two (two-component) Weyl spinors $phi$ and $chi$ which transform, correspondingly, under (½,0) and (0,½) representations of the $SO(1,3)$ group (the Lorentz group without parity transformations). Under parity transformation the Weyl spinors transform into each other.

The Dirac bispinor is connected with the Weyl bispinor by a unitary transformation to the Dirac basis,:The Dirac basis is the one most widely used in the literature.

A bilinear form of bispinors $psi^daggerotimespsi$ can be reduced to five irreducible (under the Lorentz group) objects:
# scalar, ;
# pseudo-scalar, ;
# vector, ;
# pseudo-vector, ;
# antisymmetric tensor, ,where and $\{gamma^mu,gamma\_5\}$ are the gamma matrices.

A suitable Lagrangian (the Euler-Lagrange equation of which is the Dirac equation) for the relativistic spin-½ field is given as:

Notes

References

* P. Caban and J. Rembielinski, http://arxiv.org/abs/quant-ph/0507056v1 [Phys. Rev. A 72, 012103 (2005)] .