page_title: Representation theory of the Lorentz group
section_title: The adjoint representation, the Clifford algebra, and the Dirac spinor representation
hierarchical_section_title: Representation theory of the Lorentz group / Finite-dimensional representations / The adjoint representation, the Clifford algebra, and the Dirac spinor representation
caption_attribution_description: English: German-American mathematician Richard Brauer, with his wife Ilse Brauer, in 1970
context_page_description: The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representations. This group is significant because special relativity together with quantum mechanics are the two physical theories that are most thoroughly established, and the conjunction of these two theories is the study of the infinite-dimensional unitary representations of the Lorentz group. These have both historical importance in mainstream physics, as well as connections to more speculative present-day theories.
context_section_description: In general representation theory, if (π, V) is a representation of a Lie algebra  then there is an associated representation of  on End(V), also denoted π, given by
  Likewise, a representation (Π, V) of a group G yields a representation Π on End(V) of G, still denoted Π, given by
  If π and Π are the standard representations on  and if the action is restricted to   then the two above representations are the adjoint representation of the Lie algebra and the adjoint representation of the group respectively. The corresponding representations (some  or ) always exist for any matrix Lie group, and are paramount for investigation of the representation theory in general, and for any given Lie group in particular.
  Applying this to the Lorentz group, if (Π, V) is a projective representation, then direct calculation using (G5) shows that the induced representation on End(V) is a proper representation, i.e. a representation without phase factors.
  In quantum mechanics this means that if (π, H) or (Π, H) is a representation acting on some Hilbert space H, then the corresponding induced representation acts on the set of linear operators on H. As an example, the induced representation of the projective spin (, 0) ⊕ (0, ) representation on End(H) is the non-projective 4-vector (, ) representation.
  For simplicity, consider only the "discrete part" of End(H), that is, given a basis for H, the set of constant matrices of various dimension, including possibly infinite dimensions. The induced 4-vector representation of above on this simplified End(H) has an invariant 4-dimensional subspace that is spanned by the four gamma matrices. (The metric convention is different in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime,  whose complexification is  generated by the gamma matrices decomposes as a direct sum of representation spaces of a scalar irreducible representation (irrep), the (0, 0), a pseudoscalar irrep, also the (0, 0), but with parity inversion eigenvalue −1, see the next section below, the already mentioned vector irrep, (, ), a pseudovector irrep, (, ) with parity inversion eigenvalue +1 (not −1), and a tensor irrep, (1, 0) ⊕ (0, 1). The dimensions add up to 1 + 1 + 4 + 4 + 6 = 16. In other words,
  where, as is customary, a representation is confused with its representation space.
caption: Richard Brauer and wife Ilse 1970. Brauer generalized the spin representations of Lie algebras sitting inside Clifford algebras to spin higher than 1/2. Photo courtesy of MFO.
exif: {}