Representation theory of the Lorentz group

The Lorentz group of theoretical physics has a variety of representations, corresponding to particles with integer and half-integer spins in quantum field theory. These representations are normally constructed out of spinors.

The group may also be represented in terms of a set of functions defined on the Riemann sphere. these are the Riemann P-functions, which are expressible as hypergeometric series. An important special case is the subgroup SO(3), where these reduce to the spherical harmonics, and find practical application in the theory of atomic spectra.

Finding representations

According to general representation theory of Lie groups, one first looks for the representations of the complexification of the Lie algebra of the Lorentz group. A convenient basis for the Lie algebra of the Lorentz group is given by the three generators of rotations "J"^"k"=ε^"ijk""L"_"ij" and the three generators of boosts "K"^"i"="L"_"it" where "i", "j", and "k" run over the three spatial coordinates and ε is the Levi-Civita symbol for a three dimensional spatial slice of Minkowski space. Note that the three generators of rotations transform like components of a pseudovector J and the three generators of boosts transform like components of a vector K under the adjoint action of the spatial rotation subgroup.

This motivates the following construction: first complexify, and then change basis to the components of A = (J + "i" K)/2 and B = (J – "i" K)/2. In this basis, one checks that the components of A and B satisfy separately the commutation relations of the Lie algebra sl₂ and moreover that they commute with each other. In other words, one has the isomorphism:$mathfrak\{so\}(3,1)otimesmathbb\{C\}\; cong\; mathfrak\{sl\}\_2(mathbb\{C\})oplus\; mathfrak\{sl\}\_2(mathbb\{C\}).$

The utility of this isomorphism comes from the fact that sl₂ is the complexification of the rotation algebra, and so its irreducible representations correspond to the well-known representations of the spatial rotation group; for each "j" in ½Z, one has the (2"j"+1)-dimensional spin-"j" representation spanned by the spherical harmonics with "j" as the highest weight. Thus the finite dimensional irreducible representations of the Lorentz group are simply given by an ordered pair of half-integers ("m","n") which fix representations of the subalgebra spanned by the components of A and B respectively.

The Lorentz group also has infinite dimensional unitary representations, first studied by Bargmann (1947).

Properties of the (m,n) irrep

Since the angular momentum operator is given by J = A + B, the highest weight of the rotation subrepresentation will be "m"+"n". So for example, the (1/2,1/2) representation has spin 1. The ("m","n") representation is (2"m"+1)(2"n"+1)-dimensional.

Common reps

* (0,0) the Lorentz scalar representation. This representation is carried by relativistic scalar field theories.
* (1/2,0) is the left-handed Weyl spinor and (0,1/2) is the right-handed Weyl spinor representation.
* (1/2,0) ⊕ (0,1/2) is the bispinor representation (see also Dirac spinor).
* (1/2,1/2) is the four-vector representation. The electromagnetic vector potential lives in this rep. It is a 1-form field.
* (1,0) is the self-dual 2-form field representation and (0,1) is the anti-self-dual 2-form field representation.
*(1,0) ⊕ (0,1) is the representation of a parity invariant 2-form field. The electromagnetic field tensor transforms under this representation.
*(1,1/2) ⊕ (1/2,1) is the Rarita-Schwinger field representation.
* (1,1) is the spin-2 representation of the traceless metric tensor.

Full Lorentz group

The ("m","n") representation is irreducible under the restricted Lorentz group (the identity component of the Lorentz group). When one considers the full Lorentz group, which is generated by the restricted Lorentz group along with time and parity reversal, not only is this not an irreducible representation, it is not a representation at all, unless "m"="n". The reason is that this representation is formed in terms of the sum of a vector and a pseudovector, and a parity reversal changes the sign of one, but not the other. The upshot is that a vector in the ("m","n") representation is carried into the ("n","m") representation by a parity reversal. Thus ("m","n")⊕("n","m") gives an irrep of the full Lorentz group. When constructing theories such as QED which is invariant under parity reversal, Dirac spinors may be used, while theories that do not, such as the electroweak force, must be formulated in terms of Weyl spinors.

See also

* Poincaré group
* Wigner's classification

References

* V. Bargmann, "Irreducible unitary representations of the Lorenz group". Ann. of Math. 48 (1947), 568-640.