Lorentz Group - Covering Groups

Covering Groups

In a previous section we constructed a homomorphism SL(2,C) SO+(1,3), which we called the spinor map. Since SL(2,C) is simply connected, it is the covering group of the restricted Lorentz group SO+(1,3). By restriction we obtain a homomorphism SU(2) SO(3). Here, the special unitary group SU(2), which is isomorphic to the group of unit norm quaternions, is also simply connected, so it is the covering group of the rotation group SO(3). Each of these covering maps are twofold covers in the sense that precisely two elements of the covering group map to each element of the quotient. One often says that the restricted Lorentz group and the rotation group are doubly connected. This means that the fundamental group of the each group is isomorphic to the two element cyclic group Z₂.

Warning: in applications to quantum mechanics the special linear group SL(2, C) is sometimes called the Lorentz group.

Twofold coverings are characteristic of spin groups. Indeed, in addition to the double coverings

Spin+(1,3)=SL(2,C) SO+(1,3)

Spin(3)=SU(2) SO(3)

we have the double coverings

Pin(1,3) O(1,3)

Spin(1,3) SO(1,3)

Spin+(1,2) = SU(1,1) SO(1,2)

These spinorial double coverings are all closely related to Clifford algebras.