# Representation theory of the Lorentz group

The

Lorentz group oftheoretical physics has a variety of representations, corresponding to particles with integer andhalf-integer spins inquantum field theory . These representations are normally constructed out ofspinor s.The group may also be represented in terms of a set of functions defined on the

Riemann sphere . these are theRiemann P-function s, which are expressible ashypergeometric series . An important special case is the subgroupSO(3) , where these reduce to thespherical harmonics , and find practical application in the theory ofatomic spectra .**Finding representations**According to general

representation theory ofLie group s, one first looks for the representations of thecomplexification of theLie algebra of theLorentz group . A convenient basis for the Lie algebra of the Lorentz group is given by the three generators ofrotation s "J"^{"k"}=ε^{"ijk"}"L"_{"ij"}and the three generators ofboost s "K"^{"i"}="L"_{"it"}where "i", "j", and "k" run over the three spatial coordinates and ε is theLevi-Civita symbol for a three dimensional spatial slice ofMinkowski 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 theadjoint 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**sl**_{2}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

**sl**_{2}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 ½**Z**, one has the (2"j"+1)-dimensional spin-"j" representation spanned by thespherical harmonic s 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-handedWeyl spinor and (0,1/2) is the right-handed Weyl spinor representation.

* (1/2,0) ⊕ (0,1/2) is thebispinor representation (see alsoDirac spinor ).

* (1/2,1/2) is thefour-vector representation. The electromagneticvector 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.

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