these are The choice of basis above satisfies the relations, but other choices are possible. The multiple use of the symbol above and in the sequel should be observed. For example, a typical boost and a typical rotation exponentiate as, symmetric and orthogonal, respectively. Weyl spinors and bispinors By taking, in turn, and and by setting in the general expression , and by using the trivial relations and , it follows These are the left-handed and right-handed Weyl spinor representations. They act by matrix multiplication on 2-dimensional complex vector spaces (with a choice of basis) and , whose elements and are called left- and right-handed Weyl spinors respectively. Given their direct sum as representations is formed, This is, up to a similarity transformation, the Dirac spinor representation of It acts on the 4-component elements of , called bispinors, by matrix multiplication. The representation may be obtained in a more general and basis independent way using Clifford algebras. These expressions for bispinors and Weyl spinors all extend by linearity of Lie algebras and representations to all of Expressions for the group representations are obtained by exponentiation. Open problems The classification and characterization of the representation theory of the Lorentz group was completed in 1947. But in association with the Bargmann–Wigner programme, there are yet unresolved purely mathematical problems, linked to the infinite-dimensional unitary representations. The irreducible infinite-dimensional unitary representations may have indirect relevance to physical reality in speculative modern theories since the (generalized) Lorentz group appears as the little group of the Poincaré group of spacelike vectors in higher spacetime dimension. The corresponding infinite-dimensional unitary representations of the (generalized) Poincaré group are the so-called tachyonic representations. Tachyons appear in the spectrum of bosonic strings and are associated with instability of the vacuum. Even though tachyons may not be realized in nature, these representations must be mathematically understood in order to understand string theory. This is so since tachyon states turn out to appear in superstring theories too in attempts to create realistic models. One open problem is the completion of the Bargmann–Wigner programme for the isometry group of the de Sitter spacetime . Ideally, the physical components of wave functions would be realized on the hyperboloid of radius embedded in and the corresponding covariant wave equations of the infinite-dimensional unitary representation to be known. See also Bargmann–Wigner equations Dirac algebra Gamma matrices Lorentz group Möbius transformation Poincaré group Representation theory of