Bispinor
In physics, a bispinor is an object with four complex components which transform in a specific way under Lorentz transformations: specifically, a bispinor is an element of a 4-dimensional complex vector space considered as a (½,0)⊕(0,½) representation of the Lorentz group. Bispinors are, for example, used to describe relativistic spin-½ wave functions.
In the Weyl basis, a bispinor
consists of two (two-component) Weyl spinors 
and 
which transform, correspondingly, under (½,0) and (0,½) representations of the 
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.
Expressions for Lorentz transformations of bispinors
A bispinor field 
transforms according to the rule
where 
is a Lorentz transformation. Here the coordinates of physical points are transformed according to 
, while 
, a matrix, is an element of the spinor representation (for spin 1/2) of the Lorentz group.
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